Answers
We have two expressions of permutations and combinations. We have to expand and simplify the expression:
[tex]\frac{_8P_5}{_8C_5}=\frac{P(8,5)}{C(8,5)}[/tex]We can expand the permutation as:
[tex]\begin{gathered} P(n,r)=\frac{n!}{(n-r)!} \\ P(8,5)=\frac{8!}{(8-5)!}=\frac{8!}{3!} \end{gathered}[/tex]and the combination as:
[tex]\begin{gathered} C(n,r)=\frac{n!}{r!(n-r)!} \\ C(8,5)=\frac{8!}{5!(8-3)!}=\frac{8!}{5!3!} \end{gathered}[/tex]Then, combining the two as:
[tex]\frac{P(8,5)}{C(8,5)}=\frac{\frac{8!}{3!}}{\frac{8!}{5!3!}}=\frac{1}{5!}=\frac{1}{120}[/tex]Answer: the quotient is equal to 1/120.
Related Questions
What is the x-value in the solution to the system of equations below?x = y - 52x + y = 11
Answers
x = 2
Explanation:The system of equations is:
x = y - 5............................(1)
2x + y = 11........................(2)
Substitute equation (1) into equation (2)
2(y - 5) + y = 11
2y - 10 + y = 11
Collect like terms
2y + y = 11 + 10
3y = 21
y = 21/3
y = 7
Substitute y = 7 into equation (1)
x = y - 5
x = 7 - 5
x = 2
If 3 computer disks and 5 notebooks cost $7.50 and 4 computer disks and 2 notebooks cost $6.50, how much does 1 computer disk cost?Group of answer choices$0.75
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let x be the price of the computer disc and let y be the price of the notebooks.
Given the information on the problem, we can write the following equations:
[tex]\begin{gathered} 3x+5y=7.50 \\ 4x+2y=6.50 \end{gathered}[/tex]we can solve for y on the second equation to get the following:
[tex]\begin{gathered} 4x+2y=6.50 \\ \Rightarrow2y=6.50-4x \\ \Rightarrow y=\frac{6.50}{2}-\frac{4x}{2}=3.25-2x \\ y=3.25-2x \end{gathered}[/tex]then, we can substitute this expression on the first equation to get the following:
[tex]\begin{gathered} 3x+5(3.25-2x)=7.50 \\ \Rightarrow3x+16.25-10x=7.50 \\ \Rightarrow-7x=7.50-16.25=-8.75 \\ \Rightarrow x=\frac{-8.75}{-7}=1.25 \\ x=1.25 \end{gathered}[/tex]therefore, the cost of 1 computer disk is $1.25
In a popular online role playing game, players can create detailed designs for theircharacter's "costumes," or appearance. Juan sets up a website where players can buyand sell these costumes online. Information about the number of people who visitedthe website and the number of costumes purchased in a single day is listed below.31 visitors purchased no costume.26 visitors purchased exactly one costume.2 visitors purchased more than one costume.
Answers
1) The first thing we need to do is to find the total number of visitors. This is going to be our Sample Space
31+26+2=31+28=59 Visitors altogether
2) The next thing to do is to find the favorable event. Note that we can see that 26 people purchased only one costume.
3) The final and last step is to find the probability, so we can write out the following:
[tex]P=\frac{26}{59}=0.8965\approx0.90[/tex]Thus, this is the answer: 0.90
which set of Integers is represents the length of the sides of an a isosceles triangle
Answers
Consider that atriangle is said to be an isosceles triangle if its two sides are equal.
Since the set of integers represent the length of sides, two values must be identical for an isosceles triangle.
Therefore, options C and D are ruled out, as all three integers are different in both sets.
Now,consider that any three side measures can form a triangle if the following condition is satisfied,
[tex]\text{Largest Side}<\text{ Sum of the remaining two sides}[/tex]Consider option A,
[tex]\begin{gathered} 6<6+5 \\ 6<11 \end{gathered}[/tex]As the statement holds true, the given set of integers (6,6,5) can form a triangle.
Consider option B,
[tex]\begin{gathered} 3<1+1 \\ 3<2 \end{gathered}[/tex]As the statement is false, the given set of integers (1,1,3) cannot form a triangle.
Thus, option A contains the set of values that will form a triangle whose two sides are equal, that is, an isosceles triangle.
Therefore, option A is the correct choice.
I need helpThis is from my ACT Prep guide, in the calculus portion
Answers
Given the expression:
[tex](2x^2y^3+y)^7[/tex]We will find the 4th term using the binomial theorem
the general form of the binomial theorem will be:
[tex](a+b)^n=^nC^{}_r\cdot a^{n-r}\cdot b^r[/tex]So, n = 7, r = 3, a = 2x²y³, b = y
Substitute into the general form
So, the 4th term will be:
[tex]\frac{7!}{3!4!}\cdot(2x^2y^3)^4\cdot y^3=35\cdot2^6\cdot x^8\cdot y^{12}\cdot y^3=560x^8y^{15}[/tex]So, the answer will be option 3
In right Delta*A * B * C legAB = legBC The area of the triangle is 12.5. Hypotenuse AC equals to : A. sqrt(5) B. 5sqrt(2) C. 5 D.4 sqrt 5
Answers
The are of a triangle is:
[tex]A=\frac{1}{2}b\cdot h[/tex]b is the base and h is the height
In the given triangle the base is Leg BC and the height is Leg AB:
[tex]\begin{gathered} AB=BC \\ A=12.5 \\ \\ 12.5=\frac{1}{2}AB^2 \end{gathered}[/tex]Use the equation above to solve AB:
[tex]\begin{gathered} \text{Multiply both sides of the equation by 2:} \\ 12.5\times2=2\times\frac{1}{2}AB^2 \\ \\ 25=AB^2 \\ \\ \text{Take the square root of both sides of the equation:} \\ \sqrt[]{25}=\sqrt{AB^2} \\ \\ 5=AB \end{gathered}[/tex]Then, Legs of the given right triangle have a measure of 5.
To find the measure of the hypotenuse use Pythagorean theorem:
[tex]\begin{gathered} \text{hypotenuse}^2=Leg1^2+Leg2^2 \\ \text{hypotenuse}=\sqrt[]{Leg1^2+Leg2^2} \end{gathered}[/tex][tex]\begin{gathered} \text{hypotenuse}=\sqrt[]{5^2+5^2} \\ \\ \text{hypotenuse}=\sqrt[]{25+25} \\ \\ \text{hypotenuse}=\sqrt[]{50} \end{gathered}[/tex]Find the prime factorization of 50 to simplify it:
[tex]50=2\times5\times5=2\times5^2[/tex][tex]\begin{gathered} \text{hypotenuse}=\sqrt[]{2\times5^2} \\ \\ \text{hypotenuse}=5\sqrt[]{2} \end{gathered}[/tex]After opening an ancient bottle you find on the beach, a Djinni appears. In payment for his freedom, he gives you a choice of either 50,000 gold coins or one magical gold coin. The magic coin will turn into two gold coins on the first day. The two coins will turn into four coins total at the end of two days. By the end or the third day there will be eight gold coins total. The Djinni explains that the magic coins will continue this pattern of doubling each day for one moon cycle, 28 days. Which prize do you choose?When you have made your choice, answer these questions:The number of coins on the third day will be 2×2×2. Can you write another expression using exponents for the number of coins there will be on the third day?Write an expression for the number of coins there will be on the 28th day. Is this more or less than a million coins?
Answers
Geometric Sequences
I was given two choices:
* Take a 50,000 gold coins prize, or
* Take a 1 magic coin prize that doubles every day for 28 days.
Which prize do I choose? Clearly, the second choice. But that is because I already know the numbers behind geometric sequences.
Now I explain why by answering these questions:
1. The number of coins on the third day will be 2×2×2. It can be also expressed by using exponents as follows:
[tex]2\times2\times2=2^3[/tex]The exponent of the base 2 is the number of days that have passed.
For the 28th day, the magic coin will give:
[tex]2^{28}\text{ coins}[/tex]But we need to know how many coins have accumulated since the first day. This will be the result of the sum:
[tex]S=1+2^1+2^2+2^3+2^4+\cdots+2^{28}[/tex]Given 1 equals 2 to the power of 0, we can write the sum as:
[tex]S=2^0+2^1+2^2+2^3+2^4+\cdots+2^{28}[/tex]This is the sum of a geometric sequence with a1 = 1 and a common ratio of r = 2.
The formula to compute the sum of n terms of a geometric sequence is:
[tex]S_n=a_1\cdot\frac{r^n-1}{r-1}[/tex]Since our sequence starts at n=0, there are n = 29 terms to sum:
[tex]\begin{gathered} S_{29}=1\cdot\frac{2^{29}-1}{2-1} \\ S_{29}=536,870,911 \end{gathered}[/tex]T
Michael has 26 blueberry scones and 65 raspberry scones. He wants to make as many identical bags of scones as possible. Each bag should have an equal number of blueberry scones and an equal number of raspberry scones. What is the greatest number of bags Michael can fill? Explain how you know. Each The greatest number of bags Michael can fill is bags because the GCF of 26 and 65 is bag can have blueberry and raspberry scones.
Answers
Michael has 26 blueberry scones and 65 raspberry scones.
So, to make equal number of bags , need to find GCF
Greatest common factor:
26 = 2 x 13
65 = 5 x 13
--------------------------
GCF = 13
So, Each bag should have an equal number of blueberry scones and an equal number of raspberry scones. So, each bag have 13 from both of them
As we have just 26 from blueberry scones
So, the maximum number of bags = 26/13 = 2
As a summery:
The greatest number of bags is 2 , because the gcf of 26 and 65 is 13
Each bag can have 13 blueberry and 13 raspberry scones
D. 52 3. Which statement is not true of the function? A. The independent variable is x. B. The dependent variable is y. C. When the input is 1, the output is -2. D. When the input is 8, the output is 32.
Answers
Given function
y = 5x - 8
From the function
x is an independent variable. Option A is true
y is a dependent variable. Option B is true
when the input x = 1 , y = 5x - 8
y = 5(1) - 8
y = 5 - 8
y = -3
Option C is not true
when the input x = 8 , y = 5x - 8
y = 5(8) - 8
y = 5 - 8
y = -3
what values of x make inequality 1 + 5x < -9
Answers
Dividing both sides by 5, we have
[tex]\begin{gathered} \frac{5x}{5}<\frac{-10}{5} \\ x<-2 \end{gathered}[/tex]A birdhouse is constructed of a rectangular prism with a square base and a square pyramid at the top of the house. If a student wanted to put a fence around the entire birdhouse to see all of the animals, determine how square feet of fence would be required to make the fence.
Answers
Solution:
The area of fence needed is the area of the upper pyramid plus the area of the square prism. That is:
[tex]A\text{ = Pyramid area + rectangular prism area}[/tex]the area of a Pyramid is given by the following formula:
and the area of the rectangular prism is:
A = 2lw +2lh+2wh
where
l is the lenght, h is the height and w is the width
Thus, using the data from the problem and the equations above, we can conclude that:
[tex]A\text{ = Pyramid area + rectangular prism area}=340+3200=3540[/tex]so that, the correct answer is:
[tex]3540[/tex]which expression can be used to determine the volume of the square pyramid
Answers
The expression to find the volume of the pyramid is;
[tex]V=\frac{1}{3}(\text{base}-\text{area)(height)}[/tex]This volume is, therefore;
[tex]\begin{gathered} V=\frac{1}{3}(14\times4)12 \\ V=\frac{1}{3}(14.12.4) \end{gathered}[/tex]That is Option C
I am having trouble with this problem & have no idea how to solve it.
Answers
Step 1
Given;
Step 2
[tex]\begin{gathered} f(-2)=2(-2)^2+2(-2)-1 \\ f(-2)=8-4-1=3 \end{gathered}[/tex]Answer;
[tex]f(-2)=3[/tex]Identify the algebraic property of equality. If 2x + y = 9 and y = 3, then 2x +3 = 9.
Answers
The substitution property of equality states that if x = y, then x can be substituted in for y in any equation, and y can be substituted for x in any equation.
In the case of the question, y = 3 is substituted in 2x + y = 9 to get 2x + 3 = 9
Evaluate the expression when a = 3/10 and x = -3/20 4x-a write your answer in simplest form
Answers
HELP ME PLEASEEEEEEEEEEE
Answers
Answer:
the letter of correct answer is letter C
A rectangular auditorium seats 3300 people. The number of seats in each row exceeds the number of rows by 16. Find the number of seats in eachrow.
Answers
Take x as the number of rows and y as the number of seats in each row.
There are 3300 seats in the auditorium, which means that the product of x and y is 3300:
[tex]x\cdot y=3300[/tex]We also know that y exceeds x by 16, this is:
[tex]y=x+16[/tex]Use these equations to find the number of seats and rows.
[tex]\begin{gathered} x\cdot(x+16)=3300 \\ x^2+16x=3300 \\ x^2+16x-3300=0 \end{gathered}[/tex]Use the quadratic equation to find x:
[tex]\begin{gathered} x=\frac{-16\pm\sqrt[]{16^2-4(1\cdot-3300)}}{2\cdot1} \\ x=\frac{-16\pm116}{2} \\ x1=\frac{-16+116}{2}=\frac{100}{2}=50 \\ x2=\frac{-16-116}{2}=\frac{-132}{2}=-66 \end{gathered}[/tex]As x is the number of rows, it can take negative values, so take the value of 50.
Use this values to find y:
[tex]\begin{gathered} x\cdot y=3300 \\ y=\frac{3300}{x} \\ y=\frac{3300}{50} \\ y=66 \end{gathered}[/tex]There are 50 seats in each row and there are 66 rows.
Calculate the value of x on the diagram (a) 9m2 (b) 16m2 (c) 5m (d) 3m (e) 4mpls check the attachment for the diagram
Answers
Given: A right triangle ABC with sides AB=3M and BC=4M.
Required: To determine the side length of AC.
Explanation: The side AC=X can be determined by joining B to C and considering the triangle ABC as a right-angled triangle.
Now using the Pythagoras Theorem, we can write-
[tex]AC^2=AB^2+BC^2[/tex]Substituting the values-
[tex]\begin{gathered} X^2=(3M)^2+(4M)^2 \\ X^2=9M^2+16M^2 \\ X=\sqrt{25M^2} \\ X=5M \end{gathered}[/tex]Final Answer: Option C is correct.
Amelia has 10 coins in her pocket. She has 3 nickels, 2 quarters, 3 dimes, and 2 pennies. Amelia chooses a coin at random.What is the probability that she selects a nickel or a penny from her pocket?
Answers
It is given that there are ten coins:
• 3 nickels
,• 2 quarters
,• 3 dimes
,• 2 pennies
It is required to find the probability that a nickel or a penny was selected.
Recall the formula for the probability of an event:
[tex]P=\frac{\text{ Number of favorable outcomes}}{\text{ Total number of possible outcomes}}[/tex]The total number of coins is 10. Hence, the number of possible outcomes is 10.
The number of nickels or pennies is 3+2=5. Hence, the number of favorable outcomes is 5.
Substitute these values into the probability formula:
[tex]P=\frac{5}{10}=\frac{1}{2}[/tex]The answer is 1/2.
if a spacecraft is descending at a rate of 13.81 per minute. what its change in height after 5 1/2 min.
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What is the solution to the inequality x2 > –9x – 18?
A) x < –6 or x > –3
B) –6 < x < –3
C) x ≤ –6 and x ≥ –3
D) x < –6 and x > –3
Answers
Given the inequality:
[tex]x^2>-9x-18[/tex]Solve for x as follows:
[tex]\begin{gathered} x^2+9x+18>0 \\ (x+3)(x+6)>0 \\ x+3=0\rightarrow x=-3 \\ x+6=0\rightarrow x=-6 \\ So,\text{ } \\ x>-3,x<-6 \end{gathered}[/tex]So, the answer will be option C) x < -6 and x > -3
Home work thanks need help with this problem multi part question
Answers
b) $2273.47
c) The expected salary with 13 years of experience is $78879.59
Explanation:a) To write a linear model, we will use the linear equation formula:
y = mx + b
m = slope, b = y-intercept
First we need to find the slope using the formula:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]Picking any two points on the table, we use in the formula:
[tex]\begin{gathered} (0,\text{ 49900) and (2, 52400)} \\ x_1=0,y_1=49900,x_2=2,y_2\text{ = }52400 \\ \text{slope = }\frac{52400\text{ - 49900}}{2-0} \\ \text{slope = 2500/2} \\ \text{slope = 12}50 \\ \end{gathered}[/tex]There is something wrong with the values in the table. The slope is supposed to be constant for any two points in the table but that is not so here.
I will be using a linear regression instead:
[tex]\begin{gathered} Regression\text{ }Equation\colonŷ=bX+a \\ X\text{ = years of experience} \\ ŷ\text{ = salary in dollars} \end{gathered}[/tex]Graphing the values, we have:
[tex]ŷ=2273.46939X+49324.4898[/tex]b) The slope of the model we got is 2273.47
It means the rate of salary per year of experience is $2273.47
c) for 13 years of experience, x = 13
[tex]\begin{gathered} we\text{ replace X with 13:} \\ ŷ=2273.46939(13)+49324.4898 \\ ŷ\text{ = 78879.59} \end{gathered}[/tex]The expected salary with 13 years of experience is $78879.59
Which numbers can be classified as rational? Select all that apply.A.56B. 711.C. 6.565656...D. 0.23E. 0.32416..
Answers
Answer:
A, C, D and F
Step-by-step explanation:
Non-rational numbers:
Decimal numbers in which there is not a pattern in the decimal part.
Non exact roots
In this question:
The square root of 11 is not rational, because it is not an exact root
0.32416 is not rational, because 32416 does not follow a pattern.
The others are rational.
So you should mark:
A, C, D and F
using the formula A=bh, if the base if a rectangle is 5 feet and the height of the rectangle is 10 feet,what units would represent the area?
Answers
Area of the rectangle = Base * height
A = bh, where
A is the area of the rectangle
b is its base
h is its height
In the question the base of the rectangle is 5 feet
b = 5 ft
The height of the rectangle is 10 feet
h = 10 ft
Let us substitute b and h in the formula of A
[tex]A=5\times10=50ft^2[/tex]The area of the rectangle is 50 square feet
A company that owed $2,000 paid early and got a $40 discount. What fraction of the amount owed was the discount?
Answers
A company that owed $2,000 paid early and got a $40 discount. What fraction of the amount owed was the discount?
we have that
$2,000 represent the 100%
so
Applying proportion
Find out how much percentage represent $40
100/2,000=x/40
solve for x
x=(100/2,000)*40
x=(1/20)*40
x=40/20
x=2%
2%=2/100=1/50
the fraction is 1/50Johnny your friend claims that the green trail is 10 Mi longer than the pink Trail what calculation error might Bonnie and have made what is the correct comparison of the length of the pink trail to the length of the Green trial
Answers
Johnny forgot that each little square is 1/2 of a mile. SO the correct comparison is that the green trial is 5 miles longer than the pink one
Susan Marciano invested 16,000 bonus in a fund that paid 10% profit and invested the rest in stock that suffered a 3% loss. Find the amount of each investment if her overall investment if her overran net profit was 690 The amount invested at 10% is ?The amount invested in stock is ?
Answers
The amount invested at 10% is $9000
The amount invested in stock is $7000
EXPLANATION
Let x= amount invested in the fund
y= amount invested in stock
x + y = 16 000
0.1x - 0.03y = 690
From the first equation, x = 16 000 - y
Substitute the above into the second equation
0.1(16000 - y) - 0.03y = 690
open the parenthesis
1600 - 0.1y -0.03y = 690
1600 - 0.13y = 690
Subtract 1600 from both-side of the equation
-0.13y = 690 - 1600
-0.13y = -910
Divide both-side by -0.13
y =7000
But x = 16 000 - y
x = 16 000 - 7000
x = 9000
Therefore,
The amount invested at 10% is $9000
The amount invested in stock is $7000
Could you please help me figure out this problem so I can move forward with the rest of them I figure them out
Answers
Scientific Notation has this form:
[tex]a\cdot10^b[/tex]Where the coefficient "a" is a number from 1 to 10 (10 is not included) and the exponent "b" is an Integer.
By definition, in Scientific Notation, the Decimal Point must be after the first nonzero digit of the coefficient.
In this case, you have this number written in Scientific Notation:
[tex]3.629\cdot10^7[/tex]Notice that the exponent is 7. That indicates that, in order to write it in Standard Notation, you need to move the Decimal Point 7 places to the right:
[tex]=36,290,000[/tex]Therefore, the answer is the second option:
[tex]36,290,000[/tex]A car on the highway is getting 13 kilometers per liter of gas. How many meters pe milliliter would this be? 1.3 meters per milliliter 13 meters per milliliter 26 meters per milliliter 39 meters per milliliter 24 points)
Answers
We assume that we have to convert from 13 km/lt to m/ml.
To do this conversion, we need to remember that:
1 km = 1000 m
1 lt = 1000 ml
Then, we can proceed as follows to convert those quantities:
[tex]13\cdot\frac{\operatorname{km}}{lt}\cdot\frac{1000m}{\operatorname{km}}\cdot\frac{1lt}{1000ml}=13\frac{m}{ml}[/tex]Then, the answer is 13 m/ml.
Ingrid walked her dog and washed her car. The time she spent walking her dog was one-half the time it took her to wash her car. It took Ingrid 23 minutes to walk the dog. How long did it take Ingrid to wash her car? Use the variable w to represent how long it took Ingrid to wash her car.
Answers
Let w be the time Ingrid took to wash her car.
We know that she spent half that time walking her dog, this means that:
[tex]\frac{1}{2}w[/tex]is the time she spent walking her dog. We also know that she spent 23 minutes doing that, then we have the equation:
[tex]\frac{1}{2}w=23[/tex]Solving for w we have:
[tex]\begin{gathered} \frac{1}{2}w=23 \\ w=2\cdot23 \\ w=46 \end{gathered}[/tex]Therefore she spent 46 minutes whashing the car.