Question 1

Find the numerical value.
-$$\mathrm { P } _ { 0 } ^ { 12 }$$

Accepted Answer

The notation $\mathrm{P}_0^{12}$ represents the number of permutations of 12 items taken 0 at a time, which is always 1, as there is exactly one way to arrange zero items.

Question 2

A consumer testing service is commissioned to rank the top 4 brands of dishwasher
detergent. Ten brands are to be included in the study. In how many different ways can the
consumer testing service arrive at the ranking?

Accepted Answer

The answer of A consumer testing service is commissioned to...

Question 3

A statistics quiz has 5 multiple choice questions with four answer choices each and 5
true/false questions. How many different ways can a student answer the questions by
guessing?

Accepted Answer

The answer of A statistics quiz has 5 multiple choice...

Question 4

There are 15 candidates for three different executive positions. How many different ways
could you fill the positions?

Accepted Answer

The answer of There are 15 candidates for three different...

Question 5

Suppose you teach a class of twelve students that you wish to assign to three groups of four students each. In how many different ways can groups be made?

Accepted Answer

The number of ways to divide 12 students into 3 groups of 4 is calculated by dividing the total permutations of the students by the permutations within each group and the permutations of the groups themselves: (12! / (4! * 4! * 4! * 3!)) = 34,650.

Question 6

Suppose there are four dangerous military missions, each requiring one soldier. In how
many different ways can 4 soldiers from a squadron of 100 be assigned to these four
missions?

Accepted Answer

The answer of Suppose there are four dangerous military missions,...

Question 7

Find the numerical value.
-$$P _ { 4 } ^ { 10 }$$

Accepted Answer

$$P _ { 4 } ^ { 10 } = \frac{10!}{(10-4)!} = \frac{10!}{6!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!} = 10 \times 9 \times 8 \times 7 = 5040.$$

Question 8

A panel of coaches ranks the high school football teams in a particular state each week.
The panel unanimously agrees on which five teams should comprise the top five but not
on the order. In how many different ways can the coaches rank the top five teams?

Accepted Answer

The answer of A panel of coaches ranks the high...

Question 9

Find the numerical value.
-$$\mathrm { P } _ { 4 } ^ { 8 }$$

Accepted Answer

$\mathrm { P } _ { 4 } ^ { 8 }$ refers to the number of permutations of 8 items taken 4 at a time. The formula for permutations is $\mathrm { P } _ { r } ^ { n } = \frac { n ! } { ( n - r ) ! }$. Plugging in the values, we get $\mathrm { P } _ { 4 } ^ { 8 } = \frac { 8 ! } { ( 8 - 4 ) ! } = \frac { 8 ! } { 4 ! } = \frac { 8 \times 7 \times 6 \times 5 \times 4 ! } { 4 ! } = 8 \times 7 \times 6 \times 5 = 1680$.

Question 10

A group of 13 people is traveling in three vehicles. If 6 people ride in the van, 5 people ride
in the car, and 2 people ride in the pickup, in how many different ways can the people be
assigned to the vehicles?

Accepted Answer

The answer of A group of 13 people is traveling...

Question 11

You have 4 tasks to complete and 10 people, including yourself, to complete the tasks. In how many ways can you assign 4 people to the first task, 3 people to the second task, 2 people to the
Third task, and 1 person to the fourth task?

Accepted Answer

The answer of You have 4 tasks to complete and...

Question 12

A group of 12 people is traveling in three vehicles. If 4 people ride in each vehicle, in how many different ways can the people be assigned to the vehicles?

Accepted Answer

The problem can be solved using the concept of combinations since the order of selection does not matter. We have 12 people and we want to divide them into groups of 4 to be placed in 3 vehicles. First, we select 4 people out of 12 to go into the first vehicle, then 4 out of the remaining 8 for the second vehicle, and the last 4 automatically go into the third vehicle. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose. The calculation is as follows:For the first vehicle: C(12, 4) = 12! / [4!(12-4)!] = 495 ways.For the second vehicle: C(8, 4) = 8! / [4!(8-4)!] = 70 ways.The last 4 people will automatically go into the third vehicle, so we don't need to calculate that.However, since these selections are happening simultaneously and are independent of each other, we multiply the number of ways to find the total number of combinations: 495 * 70 = 34,650.Therefore, there are 34,650 different ways the 12 people can be assigned to the 3 vehicles.

Question 13

Find the numerical value.
-$$\left( \begin{array} { l } 
7 \
2
\end{array} \right)$$

Accepted Answer

The notation $\left( \begin{array} { l } 7 \ 2 \end{array} \right)$ represents a binomial coefficient, which is calculated as $\frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21$.

Question 14

You have 5 tasks to complete and 15 people, including yourself, to complete the tasks. In
how many ways can you assign 3 people to each task?

Accepted Answer

The answer of You have 5 tasks to complete and...

Question 15

Find the numerical value.
-Find the numerical value of $P _ { 5 } ^ { 5 }$.

Accepted Answer

The answer of Find the numerical value.
-Find the numerical value...

Question 16

Find the numerical value.
-A manager of a road construction company must send one engineer to each of three different job sites to meet with government inspectors. There are 8 engineers available. How many different
Ways can the manager send one engineer to each of the job sites? (Assume that sending a particular
Engineer to job site 1 is different than sending the same engineer to either job site 2 or 3.)

Accepted Answer

This is a permutation problem because the order in which the engineers are sent to the job sites matters. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to choose, and "!" denotes factorial, which is the product of all positive integers up to that number. Here, n = 8 engineers and r = 3 job sites. So, P(8, 3) = 8! / (8-3)! = 8! / 5! = (8*7*6*5!)/5! = 8*7*6 = 336.

Question 17

Find the numerical value.
-Find the numerical value of $P _ { 4 } ^ { 6 }$.

Accepted Answer

$P _ { 4 } ^ { 6 } = \frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{720}{2} = 360.$

Question 18

Find the numerical value.
-Which expression is equivalent to $P _ { n } ^ { N } ?$

Accepted Answer

The correct expression for $P _ { n } ^ { N }$, which represents the number of permutations of $N$ items taken $n$ at a time, is $\frac { N ! } { ( N - n ) ! }$. This formula calculates the total number of ways to arrange $n$ items out of $N$ without repetition.

Question 19

Find the numerical value.
-$$\mathrm { P } _ { 4 } ^ { 5 }$$

Accepted Answer

$\mathrm { P } _ { 4 } ^ { 5 }$ represents the number of permutations of 5 items taken 4 at a time, which is calculated as $5! / (5-4)!$ = $5! / 1!$ = $5 \times 4 \times 3 \times 2 \times 1 / 1$ = 120.

Question 20

Find the numerical value.
-$\left( \begin{array} { l } 20 \ 20 \end{array} \right)$

Accepted Answer

The notation $\left( \begin{array} { l } n \ k \end{array} \right)$ represents a binomial coefficient, which is calculated as $\frac{n!}{k!(n-k)!}$. When $n = k$, the value is always 1, because $n!/(n!(n-n)!)$ simplifies to $n!/n! \cdot 0!$, and since $0! = 1$, the result is $1$.

Quiz 3: Probability

Find the numerical value. - $\mathrm { P } _ { 0 } ^ { 12 }$

A consumer testing service is commissioned to rank the top 4 brands of dishwasher detergent. Ten brands are to be included in the study. In how many different ways can the consumer testing service arrive at the ranking?

A statistics quiz has 5 multiple choice questions with four answer choices each and 5 true/false questions. How many different ways can a student answer the questions by guessing?

There are 15 candidates for three different executive positions. How many different ways could you fill the positions?

Suppose you teach a class of twelve students that you wish to assign to three groups of four students each. In how many different ways can groups be made?

Suppose there are four dangerous military missions, each requiring one soldier. In how many different ways can 4 soldiers from a squadron of 100 be assigned to these four missions?

Find the numerical value. - $P _ { 4 } ^ { 10 }$

A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously agrees on which five teams should comprise the top five but not on the order. In how many different ways can the coaches rank the top five teams?

Find the numerical value. - $\mathrm { P } _ { 4 } ^ { 8 }$

A group of 13 people is traveling in three vehicles. If 6 people ride in the van, 5 people ride in the car, and 2 people ride in the pickup, in how many different ways can the people be assigned to the vehicles?

You have 4 tasks to complete and 10 people, including yourself, to complete the tasks. In how many ways can you assign 4 people to the first task, 3 people to the second task, 2 people to the Third task, and 1 person to the fourth task?

A group of 12 people is traveling in three vehicles. If 4 people ride in each vehicle, in how many different ways can the people be assigned to the vehicles?

Find the numerical value. - $\left( \begin{array} { l } 7 \\2\end{array} \right)$

You have 5 tasks to complete and 15 people, including yourself, to complete the tasks. In how many ways can you assign 3 people to each task?

Find the numerical value. -Find the numerical value of $P _ { 5 } ^ { 5 }$ .

Find the numerical value. -Find the numerical value of $P _ { 4 } ^ { 6 }$ .

Find the numerical value. -Which expression is equivalent to $P _ { n } ^ { N } ?$

Find the numerical value. - $\mathrm { P } _ { 4 } ^ { 5 }$

Find the numerical value. - $\left( \begin{array} { l } 20 \\ 20 \end{array} \right)$

Quiz 3: Probability

Find the numerical value. - P012\mathrm { P } _ { 0 } ^ { 12 }P012​

A consumer testing service is commissioned to rank the top 4 brands of dishwasher detergent. Ten brands are to be included in the study. In how many different ways can the consumer testing service arrive at the ranking?

A statistics quiz has 5 multiple choice questions with four answer choices each and 5 true/false questions. How many different ways can a student answer the questions by guessing?

There are 15 candidates for three different executive positions. How many different ways could you fill the positions?

Suppose you teach a class of twelve students that you wish to assign to three groups of four students each. In how many different ways can groups be made?

Suppose there are four dangerous military missions, each requiring one soldier. In how many different ways can 4 soldiers from a squadron of 100 be assigned to these four missions?

Find the numerical value. - P410P _ { 4 } ^ { 10 }P410​

A panel of coaches ranks the high school football teams in a particular state each week. The panel unanimously agrees on which five teams should comprise the top five but not on the order. In how many different ways can the coaches rank the top five teams?

Find the numerical value. - P48\mathrm { P } _ { 4 } ^ { 8 }P48​

A group of 13 people is traveling in three vehicles. If 6 people ride in the van, 5 people ride in the car, and 2 people ride in the pickup, in how many different ways can the people be assigned to the vehicles?

You have 4 tasks to complete and 10 people, including yourself, to complete the tasks. In how many ways can you assign 4 people to the first task, 3 people to the second task, 2 people to the Third task, and 1 person to the fourth task?

A group of 12 people is traveling in three vehicles. If 4 people ride in each vehicle, in how many different ways can the people be assigned to the vehicles?

Find the numerical value. - (72) \left( \begin{array} { l } 7 \\2\end{array} \right) (72​)

You have 5 tasks to complete and 15 people, including yourself, to complete the tasks. In how many ways can you assign 3 people to each task?

Find the numerical value. -Find the numerical value of P55P _ { 5 } ^ { 5 }P55​ .

Find the numerical value. -Find the numerical value of P46P _ { 4 } ^ { 6 }P46​ .

Find the numerical value. -Which expression is equivalent to PnN?P _ { n } ^ { N } ?PnN​?

Find the numerical value. - P45\mathrm { P } _ { 4 } ^ { 5 }P45​

Find the numerical value. - (2020) \left( \begin{array} { l } 20 \\ 20 \end{array} \right) (2020​)

Find the numerical value. - $\mathrm { P } _ { 0 } ^ { 12 }$

Find the numerical value. - $P _ { 4 } ^ { 10 }$

Find the numerical value. - $\mathrm { P } _ { 4 } ^ { 8 }$

Find the numerical value. - $\left( \begin{array} { l } 7 \\2\end{array} \right)$

Find the numerical value. -Find the numerical value of $P _ { 5 } ^ { 5 }$ .

Find the numerical value. -Find the numerical value of $P _ { 4 } ^ { 6 }$ .

Find the numerical value. -Which expression is equivalent to $P _ { n } ^ { N } ?$

Find the numerical value. - $\mathrm { P } _ { 4 } ^ { 5 }$

Find the numerical value. - $\left( \begin{array} { l } 20 \\ 20 \end{array} \right)$