Answers
Answer:
6.2
Step-by-step explanation:
Formula :
Distance between two points = √(xB−xA)2+(yB−yA)2(xB-xA)2+(yB-yA)2
Solution :
Distance between two points = √(5.5−1)2+(9.25−5)2(5.5-1)2+(9.25-5)2
= √4.52+4.2524.52+4.252
= √20.25+18.062520.25+18.0625
= √38.312538.3125 = 6.1897
Related Questions
part a: find the right reimann sum estimate of the integral from 1 to 5 of f of x dx, based on the subintervals given in the table. (4 points)
Answers
A right Riemann sum has the formula A=∑ni=1Δxf(xi) A = ∑ i = 1 n Δ x f ( x i ) where x is the width of each of the n rectangles and f(xi) is the height.
what is is Riemann sum?A specific method of approximating an integral with a finite sum in mathematics is known as a Riemann sum. Riemann is the name of a German mathematician who lived in the 19th century. Approximating the area of functions or lines in graphs, as well as the length of curves and other approximations, is a highly typical application. The picked points in each subinterval of the partition are all in the upper Riemann sum, given an interval partition. The interval is often separated into subintervals of equal size.
Riemann sum formula is A=∑f(xi)Δx A = ∑ f ( x i ) Δ x ,
where A is the curve's area under the evaluable interval,
f(xi) f ( x i ) is each rectangle's height (or the average of the two heights in the case of a trapezoid)
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felipe graphs the linear relationship passing through the points 5,11 and -3,-9
Answers
slope -20/-8
line y = -5/2x - 3/2
it is NOT proportional because it doesnt pass thru 0,0
URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
Answers
The length of diameter of the base of the cone will be 112 cm. Hence, option b is correct.
What is the Pythagoras Theorem in Math?The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.Pythagoras, a Greek philosopher who was born around 570 BC, is remembered by the theorem's name. The theorem has likely been proved the most times of any mathematical theorem using a variety of techniques. The proofs are many, some of which go back thousands of years, and include both geometric and algebraic proofs.Given data :
As per the given information in the question,
Use the equation of Pythagoras's theorem,
106² = 90² + r²
r² = 106² - 90²
r = √3136
r = 56 cm.
Then, the diameter will be,
D = 2r
D = 2(56)
D = 112 cm.
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7=y/3
Solve and create new equation
Answers
Y=21
Mark me Brainly
f(x) = 3x^2 + 5
g(x) = 4x - 2
h(x) = x^2 - 3x + 1
Find f(x) + g(x) - h(x).
Answers
The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
The functions are given below.
f(x) = 3x² + 5
g(x) = 4x - 2
h(x) = x² - 3x + 1
The expression is given as,
⇒ f(x) + g(x) - h(x)
⇒ (3x² + 5) + (4x - 2) - (x² - 3x + 1)
⇒ 3x² + 5 + 4x - 2 - x² + 3x - 1
⇒ 2x² + 7x + 2
The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.
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The following questions are about a spherical balloon that is being filled with air such that its radius is increasing at a constant rate of 2 cm/sec.
Part A:
How fast is the surface area increasing when the radius of the sphere is 10 cm? Round to the nearest thousandths.
Part B:
How fast is the volume increasing when the radius is 10 cm? Round to the nearest thousandths.
Answers
The surface area increases when the radius of the sphere is 160π cm^2/s
Given that dt/dr =10 cm/s and r=15 cm
Find out dt/ds=?
We know that the surface area of sphere S=4πr^2
dt/ds=4π(2r)
dt/dr=8π(2)(10)
=160π cm^2/s.
Let r, V, and S be respectively the radius, volume, and surface area of the spherical balloon at time t.
∴V=4/3 πr^3 and S=4πr^2
Rate of change of surface area with respect to t
⇒ dt/dS=2
⇒ d/dt=(4πr^2)=2
⇒8πr dt/dr =24
⇒ dr/dt=1/4πr.... (i)
Rate of change of volume of the balloon with respect to t
=dV/dt
=d/dt(4/3πr^3)
= 4/3π⋅3r^2dr/dt
=4πr ^2⋅ dr/dt
=4πr^2.1/4πr ....[From (i)]
r=10
Hence, the volume of the spherical balloon is increasing at the rate of 10cm^3/sec.
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Given m is parallel to n, find the value of x
Answers
Answer:
x = 44
Step-by-step explanation:
Given that line m is parallel to line n, the angles that we see are same-side exterior angles, meaning that these two angles add up to 180°.
Set up the equation: [tex]112 + (2x - 20) = 180[/tex] and solve for x.
. . . . . . . . . . . . 112 - 112 + 2x - 20 = 180 - 112
. . . . . . . . . . . . 2x - 20 + 20 = 68 +20
. . . . . . . . . . . . 2x /2 = 88 /2
. . . . . . . . . . . . ∴ x = 44
let v be a real vector space. suppose that v1, v2, v3, v4 are vectors in v which are linearly independent. show that the vectors v1, v1 v2, v1 v2 v3, v1 v2 v3 v4 are also linearly independent.
Answers
given that v1,v2 ,v3,v4 is linearly independent vector so in order for a set of vectors to be linearly independent it must satisfy the below conditon:
[tex]x_1v1+x_2v2+x_3v3+x_4v4[/tex] =0 --(i)
where [tex]x_1, x_2,x_3 ,x_4[/tex] is some real numbers.
a) so in order to show that v1,v2+v4,v3 is linearly independent we have to
check if equations formed by these vectors can be transformed into some equation of equation (i) form.
so we have :
[tex]x_1v1+x_2(v2+v4) +x_3v3[/tex] ---(ii)
where [tex]x_1, x_2,x_3[/tex] is some real numbers.
after evaluating equation (ii) we have:
[tex]x_1v1+x_2v2+x_3v3+x_2v4[/tex]
so the above equation is now transformed in the form of equation (i) where [tex]x_2 = x_4[/tex]
and hence [tex]x_1v1+x_2v2+x_3v3+x_2v4[/tex] =0 ----(iii)
equation (iii) shows that the set of given vectors are linearly independent.
b) so in order to show that v1 +v2, v3, v4 is linearly independent we have to check if equations formed by these vectors can be transformed into some equation of equation (i) form.
so we have this equation:
[tex]x_1(v1+v2)+ x_2v3 + x_3v4[/tex] where [tex]x_1, x_2,x_3[/tex] is some real numbers.
after evaluating equation (iv) we have:
[tex]x_1v1+x_1v2+x_2v3+x_3v4[/tex]
so the above equation is now transformed in the form of equation (i) where the cofficient of v2 is also [tex]x_1[/tex]
and hence [tex]x_1v1+x_1v2+x_2v3+x_3v4[/tex] =0 ----(v)
equation (v) shows that the set of given vectors are linearly independent.
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Complete question :
let v be a real vector space. suppose that v1, v2, v3, v4 are vectors in v which are linearly independent. show that the vectors a) v1, v2+v4, v3 and b) v1 +v2, v3, v4 are also linearly independent.
The vectors v1, v1+v2, v1+v2+v3, v1+v2+v3+v4 are also linearly independent is hence proved.
let v be a real vector space.
The v1, v2, v3, v4 are vectors in v which are linearly independent.
V={v1, v2, v3, v4} then set V is called linearly independent if [tex]\forall \alpha_i \in F \\[/tex].
∝1v1+∝2v2+∝3v3+..........+∝nvn=0
then, [tex]\forall \alpha_i =0 \\[/tex] i=1,2,3......n.
let ∝1,∝2,∝3,∝4 ε F
∝1v1+∝2(v1+v2)+∝3(v1+v2+v3)+∝4(v1+v2+v3+v4)=0
v1(∝1+∝2+∝3+∝4)+v2(∝2+∝3+∝4)+v3(∝3+∝4)+v4.∝4=0
Since v1, v2, v3, v4 are linearly independent vectors.
So ∝1+∝2+∝3+∝4=0......(1)
∝2+∝3+∝4=0....(2)
∝3+∝4=0....(3)
∝4=0...(4)
Solving eq(1), (2), (3), (4)
we get ∝1=∝2=∝3=∝4=0
So according to definition of linearly independent vectors.
A set of vectors is called linearly independent if no vector in the set can be expressed as a linear combination of the other vectors in the set. If any of the vectors can be expressed as a linear combination of the others, then the set is said to be linearly dependent.
So v1, v1+v2, v1+v2+v3, v1+v2+v3+v4 are also linearly independent vectors hence proved.
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Socrates was alive from 469 to 399 B.C.E , Plato from 427 to 347 B.C.E ,Aristotle from 384 to 322 B.C.E. What years were they all alive simultaneously
Answers
Least=322bce
Answer= anytime when the aristotles were alive
can someone help? i don't understand cos at all
Answers
Cos W as a fraction in the simplest terms is 2/3
Trigonometric ratios in a Right angled triangle:Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle.
From a Right-angled triangle with an angle θ
sin θ = Opposite side/Hypotenuse
cos θ = Adjacent side/Hypotenuse
tan θ = Opposite side /Adjacent side
Here we have
A triangle UVW
where VW = 10 units
And WU = 15 units
As we know
cos θ = Adjacent side / Hypotenuse
cos W = VW/WU = 10/15
cos W = 2/3 [ divided by 5 ]
Therefore,
Cos W as a fraction in the simplest terms is 2/3
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Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug’s effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient’s can be given by the function C(t)=-5t^2+10t, where the time, t, is hours after injection.
Part A : What are domain and range of the function C(t) based on the context of the problem?
Part B : Graph the function to determine the greatest concentration of the medication that a patient will have in their body
Answers
The domain and range of the function are (-∞,∞) and (-∞,5). The greatest concentration will be 5.
What is the range and domain of a function?The set of all values that a function can accept is known as its range, while the set of all values for which a function is defined is known as its domain.
The dependent variable's range is its domain, which is for the independent variable.
Part (A)
As per the given function C(t) = - 5t² + 10t
C'(t) = 0
-10t + 10 = 0
10t = 10
t = 1
C''(t) = -10 < 0
So, at t = 1 the function will be maximum.
C(1) = -5(1)² + 10 x 1 = 5
Thus, the range will be (-∞,5) and the domain will be (-∞,∞).
The range represents the amount of concentration while the domain is the time.
Part (B)
The graph of the function has been drawn below.
At t = 1 the C(t) is maximum as 5 thus it will be the maximum concentration.
Hence"The function's domain and range are (-∞,∞) and (-∞,5), respectively. Five will have the highest concentration.".
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Jacob is traveling down a level roadway with a speed of 9.5 m/s. He slams on the breaks and encounters a coefficient of friction of 0.847. What distance does his 985 kg truck skid before stopping?
Answers
To find the distance that Jacob's truck skids before coming to a stop, we can use the formula for the distance traveled while braking, which is:
distance = initial velocity^2 / (2 * coefficient of friction * acceleration due to gravity)
In this case, the initial velocity is 9.5 m/s, the coefficient of friction is 0.847, and the acceleration due to gravity is 9.8 m/s^2. Plugging these values into the formula, we get:
distance = (9.5 m/s)^2 / (2 * 0.847 * 9.8 m/s^2)
Solving for distance, we get:
distance = 8.76 meters
Therefore, Jacob's truck will skid a distance of 8.76 meters before coming to a stop.
The ratio of the number of model cars that Jim owns to the number of model cars Terrence owns is 8 : 6. Terrence owns 36 models cars. How many model cars does Jim own?
Answers
The number of Jim's model car is 48
How to determine the number of JIm's carFrom the question, we have the following parameters that can be used in our computation:
Ratio of the number of model cars that Jim owns to the number of model cars Terrence owns is 8 : 6
This means that
Jim : Terrence = 8 : 6
Also from the question, we have
Terrence owns 36 models cars
This means that
Terrence = 36
Substitute the known values in the above equation, so, we have the following representation
Jim : 36 = 8 : 6
Multiply by 6
Jim : 36 = 48 : 36
So, we have
Jim = 48
Hence, the number of car is 48
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Cai drove 189 miles in 7 hours. If he continued at the same rate, how far would he travel in 12 hours?
Answers
189 divided by 7 = 27
27 multiple by 12 = 324
324 Miles
If you divide 189 by 7, you get 27. If you then multiply 27(miles per hour) by 12(hours) you get 324.
A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner is divided into two equal sections labeled A and B. Nigel will spin each spinner one time.
How many of the possible outcomes have a number less than 3 or an A?
Hint: first write out the sample space (you should have 10)
Which samples have a number less than 3 or an A?
Answers
9/10 is the possible outcome that has a number less than 3 or an A.
What is probability?
The proportion of possible outcomes to outcomes in an exhaustive set of equally likely alternatives that result in a certain event.
Here, we have
A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner is divided into two equal sections labeled A and B.
We have to find how many of the possible outcomes have a number less than 3 or an A.
Denote A the first event and B the second event:
P(A∪B) = P(A) + P(B)
A = Get a number less than 3
P(A) = 2/5
B = Get section A
P(B) = 1/2
Then,
P(A∪B) = 2/5 + 1/2
P(A∪B) = 9/10
Hence, 9/10 is the possible outcome that has a number less than 3 or an A.
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Write the equation of the line that is perpendicular to the x-axis and passes through the point (6,-2).
Answers
The slope of a line that is perpendicular to the x-axis is undefined, so we cannot use the standard form of the equation of a line, y = mx + b, where m is the slope of the line. Instead, we can use the point-slope form of the equation of a line, which is given by y - y1 = m(x - x1). In this case, m is undefined, so we can simplify the equation to y = x1 - y1, where x1 and y1 are the coordinates of the point that the line passes through. In our case, this gives us the following equation:
y = 6 - (-2)
= 8
Therefore, the equation of the line that is perpendicular to the x-axis and passes through the point (6,-2) is y = 8.
(2x4-2x³ +7+ 8x²-x) / (x²+2)
Answers
The Alternative form of the ''(2x4-2x³ +7+ 8x²-x) / (x²+2)''
is 2[tex]x^{2}[/tex] - 2x + 4 + [tex]\frac{3x - 1}{x^{2} + 2}[/tex]
How to solve algebra equation ?One of the many branches of mathematics is algebra. Algebra, which is a common thread throughout practically all of mathematics, is broadly defined as the study of mathematical symbols and the rules for manipulating these symbols in formulae.One-step equations can be solved in four different ways: by adding, subtracting, multiplying, and dividing. In an equation, both sides will stay equal if we add the same amount to either side.The addition rule and the multiplication/division rule are the two methods for solving equations that are introduced to us in algebra 1. According to the equation addition rule, an equation's solution set can have the same amount added to both sides without altering.Find the attachment answer
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Which statement is missing in the proof?
Answers
The correct answer is m√5 + m√2 = 180°.
What are parallel lines?Parallel lines are those in a plane that are regularly spaced apart from one another. There is no crossing of parallel lines.
Given, k and p are parallel lines.
In an expression or equation, one value can be substituted for another and the result will still be the same. This is known as the substitution property of equality. If the values of x and y are the same, then x and y can be substituted for one another.
And we have,
m√1 + m√2 = 180°
And m√5 = m√1
So, m√5 + m√2 = 180°
Therefore, in the proof, the missing term is m√5 + m√2 = 180°.
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Divide 8.88 divided by 10 . Explain how you determine the placement of the decimal point in your quotient and how that changed the value of each digit
Answers
With the decimal point in the proper place, dividing decimals is equivalent to dividing entire numbers.
How to Divide Decimals?While dividing decimals, we must remember to place the decimal point correctly in the quotient. This is similar to how we divide whole numbers. The decimal point, which separates a decimal number's whole number and fractional parts, is a dot. A value less than 1 can be found in the digits following the decimal point. 24 is a good example. A decimal number, 15, has a fractional component of 15 and a whole number component of 24. There are two situations in which dividing decimal numbers might occur: the first calls for dividing decimals by whole numbers, and the second requires dividing decimals by decimals.To Learn more About dividing decimals refer To:
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The function g is defined as follows for the domain given.
g(x)=2x-2, domain = {-3, -2, 1, 5)
Write the range of g using set notation. Then graph g.
Answers
Answer:
1. range = {-8,-6,0,8}
2. My graph is attached to this response as a photo.
Step-by-step explanation:
1. ⭐What is the "domain" of a function?
Domain is another word for the x-values⭐What is the "range" of a function?
Range is another word for the y-values⭐What is the "set notation"?
Set notation is a way you can list numbers, such as x-values and y-values. When writing in set notation, you must list the numbers from least to greatest.Therefore, we have to find the corresponding y-values for the given x-values, and then write said y-values in set notation.
To find the corresponding y-values for the given x-values, substitute each given x-value into g(x) and solve for g(x).
✦ When x = -3
[tex]g(x) = 2x-2[/tex]
[tex]g(-3) = 2(-3) -2[/tex]
[tex]g(-3) = -6 -2[/tex]
[tex]g(x) = -8[/tex]
✧ y = -8
________________________________________________________
✦ When x = -2
[tex]g(-2) = 2(-2) -2[/tex]
[tex]g(-2) = -4-2[/tex]
[tex]g(-2) = -6[/tex]
✧ y = -6
________________________________________________________
✦ When x = 1
[tex]g(1) = 2(1) -2[/tex]
[tex]g(1) = 2-2[/tex]
[tex]g(1) = 0[/tex]
✧ y = 0
________________________________________________________
✦ When x = 5
[tex]g(5) = 2(5) -2[/tex]
[tex]g(5) = 10 -2[/tex]
[tex]g(5) = 8[/tex]
✧ y = 8
________________________________________________________
Next, we have to write the y-values in set notation. Make sure to write it from least to greatest.
range = {-8, -6, 0, 8}
An amusement park charges an admission fee of 40 dollars per person. The cost, C (in dollars), of admission for a group of p people is given by the following.
C=40 p
What is the cost of admission for a group of 4 people?
_____dollars
Answers
The cost of admission for a group of 4 people is $160
How to determine the cost of admission for a group of 4 people?From the question, we have the following parameters that can be used in our computation:
C = 40 p
Where p is the number of people
For a group of 4 people, we have
p = 4
Substitute p = 4 in the equation C = 40 p
So, we have the following representation
C(4) = 40 * 4
Evaluate the products
C(4) = 160
hence, the cost is $160
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An elephant weighed 14,000 pounds. The handlers realized the
elephant was overweight and put it on a special meal plan. The
elephant lost 2,100 pounds. What was the percent of decrease in its
body weight?
14000
Answers
The value of the decrease percentage of elephant weight is 15%.
What is percentage?A percentage is a value that indicates 100th part of any quantity. A percentage can be converted into a fraction or a decimal by dividing it by 100.
The original weight of elephant is 14000 pounds.
And, the drop in weight is 2100 pounds.
Now, the percent value of the drop in weight is calculated as follows
Percent drop = Drop in weight / Original weight × 100
⇒ 2100/14000 × 100
⇒ 15%
Hence, the percent of decrease in weight is given as 15%.
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1.
What is the area of this figure?
Answers
Answer:
44 unit^2.
Step-by-step explanation:
Divide the figure into 2 trlangles, a rectangle and a trapezium
Area = 2 * 1/2 *4 * 3 + 2 ** + 1/2 * 8 * (1 + 3)
= 12 + 16 + 16
= 44.
Which linear function represents the table?
X
4
-2
1
8
-3
y
26
-10
8
50
-16
y = 6x + 2
Oy=6x-2
y=-6x + 2
y=-6x-2
Answers
From the table, you can see that the y-values increase by 6 for every increase of 1 in the x-values. Therefore, the slope of the line is m = 6. The y-intercept is the y-value when x = 0, which is 8 according to the table. Therefore, the y-intercept is b = 8.
Substituting these values into the slope-intercept form, you get:
y = mx + b = 6x + 8
This means that the linear function that represents the table is y = 6x + 8. The other options given in the question are not correct because they do not have a slope of 6 or a y-intercept of 8.
Answer:
A.) y=6x+2
Step-by-step explanation:
I got this answer by first finding the slope. To find the slope you use the slope formula which is y2-y1 over x2-x1. When I set up the equation I got -10-26 over -2-4 and I got -36/-6 which is 6. So your slope is 6. So then you would mark out answer c and d because their slope is negative and the slope I got is positive. To find the y-int I put in two points online and I put in the slope. I did this twice with four different points just to make sure that it was right and then it gave me that the y-int is 2. So then you would mark out b because the y-int is not negative but positive. (Hope this helps in some way...if not then I'm sorry..)
Shaina is 55 kg. What is Shaina's weight in grams
Answers
Answer: 55000 grams
Step-by-step explanation:
55 kg = x grams
Cross multiplying, we get:
(1)(x) = (1000)(55)
x = 55000 grams
Determine the inverse of the function f(x) = log2(3x + 4) − 5.
f inverse of x is equal to 2 to the power of x plus 3 all over 2
f inverse of x is equal to the quantity x plus 5 end quantity squared minus 4 all over 3
f inverse of x is equal to 2 to the power of the quantity x plus 5 end quantity minus 4 all over 3
f inverse of x is equal to 2 to the power of the quantity x minus 5 end quantity plus 4 all over 3
Answers
The inverse function of the function f(x) = log₂(3x + 4) − 5 is y = [2^(x + 5) - 4]/3
How to determine the inverse of the function?From the question, we have the following equation that can be used in our computation:
f(x) = log2(3x + 4) − 5.
Rewrite as
f(x) = log₂(3x + 4) − 5.
Express f(x) as y
This gives
y = log₂(3x + 4) − 5.
Swap the positions of x and y
So, we have the following representation
x = log₂(3y + 4) − 5.
Add 5 to both sides
log₂(3y + 4) = x + 5
So, we have
3y + 4 = 2^(x + 5)
Subtract 4 from both sides
3y = 2^(x + 5) - 4
Divide b y3
y = [2^(x + 5) - 4]/3
Hence, the inverse function is y = [2^(x + 5) - 4]/3
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Please help me!!!!!!! You guys are my only hope!!! Correct answer gets brainliest
Answers
URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
Answers
The length of diameter of the base of the cone will be 112 cm.
What is meant by an equation?A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.
A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign.
Equations are mathematical expressions with two algebras flanking the equal (=) sign on either side. This relationship is illustrated by the left and right expressions being equal to one another. Left-hand side equals right-hand side is a basic, straightforward equation.
By using Pythagoras's theorem,
106² = 90² + r²
simplifying the above equation, we get
r² = 106² - 90²
r = √3136
radius = 56 cm.
Then, the diameter will be,
Diameter = 2 × radius
substitute the values in the above equation,
D = 2(56)
D = 112 cm.
Therefore, the correct answer is option b. 112 cm.
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Graph the function y =2x + 5, using a table.
Answers
The values are satisfied by
X -5 -4 -3 -2 -1
Y -5 -3 -1 1 3
How do you graph a function?Defining a function's graph The collection of all points in the plane of the form (x, f(x)) that make up a function f's graph. We may alternatively say that the graph of f is the graph of y = f. (x). As a result, the graph of an equation is a particular instance of the graph of a function.You must choose x-values and insert them into the equation in order to graph a function. You will obtain a y-value if you enter those numbers into the equation. Your coordinates for a single point are made up of your x-values and your y-values.Given data :
Y= 2X +5
let us put the value for X
X= 0
Y = 5
and, X=1
Y = 2 + 5 = 7
and, X= 2
Y= 4 + 5 = 9
and, X=3
Y = 6 + 5 = 11
and, X= 5
Y = 10 + 5= 15
and, X= -1
Y= -2 + 5 = 3
Hence, these values are satisfied by
x y
-5 -5
-4 -3
-3 -1
-2 1
-1 3
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en singers have signed up for a signing competition. The order in which they will compete is selected by random draw. Five of the signers will compete before an intermission and five will compete after the intermission. How many different ways are there for the five singers to be selected to sing before intermission? a. 40 b. 100,000 a. с. 252 d. 30,240
Answers
As a result, the singers can finish first, second, third, fourth, or fifth in 524,160 different ways.
Permutation, where order is important:
_nP_{r}=\dfrac{n!}{(n-r)!}
P r = (n−r)!n!
Combination of definitions (order is irrelevant):
_nC_{r}=\dfrac{n!}{r!(n-r)!}n C r equals r!(nr)!n!
with n!=n, cdot (n-1) ..., cdot 2..., cdot 1n!=n(n1)...2,1.
We want to know how many different ways we can choose five people from a group of 16 (ranked first, second, third, fourth, and fifth).
n=16 n=16 r=5 r=5 Because we are interested in the first, second, third, fourth, and fifth places, order is important (changing the first and second places results in different rankings and prizes for the people), so we need to use a "textbf permutation" permutation.
Analyze the permutation definition:
"Begin align*" means "16P_5" and "dfrac" 16! color#8895cetextDefinition combination &=dfrac16!11!&color#8895cetext Evaluate difference &=dfrac16cdot 15cdot 14cdot 13cdot 12cdot 11cdot 10cdot...cdot 111cdot 10...cdot 1
16!= 11!
16! = 11⋅10⋅...⋅1
16⋅15⋅14⋅13⋅12⋅11⋅10⋅...⋅1
=16⋅15⋅14⋅13⋅12
=524,160
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Full Question = There are 16 finalists in a singing competition. The top five singers receive prizes. How many ways can the singers finish first through fifth?