Question 1

Solve the problem.
-A license plate is to consist of 3 letters followed by 5 digits. Determine the number of different License plates possible if the first letter must be an N , M , or P and repetition of letters and Numbers is not permitted.

Accepted Answer

The answer of Solve the problem.
-A license plate is to...

Question 2

Suppose a traveler wanted to visit a museum, an art gallery, and the state capitol building. 45 -minute tours are offered at each attraction hourly from 10 a.m. through 3 p.m. (6 different hours). Solve the problem, disregarding travel time .
-In how many ways could the traveler schedule all three tours before 1 p.m.?

Accepted Answer

The traveler can visit the three attractions in any order, but all tours must be completed by 1 p.m. This means the tours must be scheduled at 10 a.m., 11 a.m., and 12 p.m. There are 3! (3 factorial) ways to arrange the tours, which is 6.

Question 3

Evaluate the expression.
-$${ } _ { 8 } C _ { 0 }$$

Accepted Answer

$${ } _ { 8 } C _ { 0 }$$ represents the number of ways to choose 0 items out of 8, which is always 1, regardless of the number of items.

Question 4

Suppose a traveler wanted to visit a museum, an art gallery, and the state capitol building. 45 -minute tours are offered at each attraction hourly from 10 a.m. through 3 p.m. (6 different hours). Solve the problem, disregarding travel time .
-In how many ways could the traveler schedule two of the three tours in one day?

Accepted Answer

The answer of Suppose a traveler wanted to visit a...

Question 5

Solve the problem.
-There are 9 horses in a race. In how many ways can the first three positions of the order of the Finish occur? (Assume there are no ties.)

Accepted Answer

This is a permutation problem because the order in which the horses finish matters. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items, and r is the number of items being chosen. Here, n = 9 (horses) and r = 3 (positions). So, P(9, 3) = 9! / (9-3)! = 9! / 6! = (9×8×7×6!)/6! = 9×8×7 = 504.

Question 6

Suppose a traveler wanted to visit a museum, an art gallery, and the state capitol building. 45 -minute tours are offered at each attraction hourly from 10 a.m. through 3 p.m. (6 different hours). Solve the problem, disregarding travel time .
-In how many ways could the traveler schedule all three tours in one day, with the museum tour Being after noon?

Accepted Answer

The museum tour can be scheduled at 1 p.m., 2 p.m., or 3 p.m. (3 options). The art gallery and state capitol tours can be scheduled in any of the remaining 5 time slots. The number of ways to schedule these two tours is 5P2 = 20. Therefore, the total number of ways is 3 * 20 = 60.

Question 7

Evaluate the expression.
-$${ } _ { 7 } \mathrm { C } _ { 4 }$$

Accepted Answer

$${ } _ { 7 } \mathrm { C } _ { 4 } = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.$$

Question 8

Solve the problem.
-In how many ways can the letters in the word PAYMENT be arranged if the letters are taken 4 at a Time?

Accepted Answer

The word PAYMENT has 7 distinct letters. The number of ways to arrange 4 letters out of 7 is calculated using the formula for permutations: P(n, r) = n! / (n-r)!, where n is the total number of items, and r is the number of items to be arranged. Here, n=7 and r=4, so P(7, 4) = 7! / (7-4)! = 7! / 3! = (7×6×5×4) / (3×2×1) = 840.

Question 9

Solve the problem.
-How many ways can a president, vice-president, secretary, and treasurer be chosen from a club With 9 members? Assume that no member can hold more than one office.

Accepted Answer

This is a permutation problem because the order in which the officers are chosen matters. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items, and r is the number of items to choose. Here, n=9 (members) and r=4 (offices), so the calculation is 9P4 = 9! / (9-4)! = 9! / 5! = (9*8*7*6*5!)/5! = 9*8*7*6 = 3024.

Question 10

Solve the problem.
-How many different three-digit numbers can be written using digits from the set {2, 3, 4, 5, 6} Without any repeating digits?

Accepted Answer

For a three-digit number without repeating digits using the set {2, 3, 4, 5, 6}, the first digit can be any of the 5 options (since it can't be 0 and we have 5 numbers to choose from), the second digit can be any of the remaining 4 options (since the first digit has been chosen and cannot be repeated), and the third digit can be any of the remaining 3 options. Thus, the total number of different three-digit numbers is 5 * 4 * 3 = 60.

Question 11

Solve the problem.
-How many different three-number ʺcombinationsʺ are possible on a combination lock having 22 Numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation Locks.)

Accepted Answer

The total number of different three-number combinations (permutations, since order matters) possible on a combination lock with 22 numbers, assuming no repeats, is calculated using the permutation formula P(n, r) = n! / (n-r)!, where n is the total number of options (22) and r is the number of selections (3). Thus, P(22, 3) = 22! / (22-3)! = 22! / 19! = 22 x 21 x 20 = 9240.

Question 12

Solve the problem.
-A license plate is to consist of 2 letters followed by 4 digits. Determine the number of different License plates possible if repetition of letters and numbers is permitted.

Accepted Answer

Since repetition is allowed, there are 26 choices for each of the 2 letters and 10 choices for each of the 4 digits. Thus, the total number of different license plates possible is $26^2 \times 10^4 = 676,000$.

Question 13

Evaluate the expression.
-$\frac { { } _ { 7 } C _ { 4 } } { { } _ { 7 } C _ { 2 } }$

Accepted Answer

The expression simplifies to $\frac{{7! / (4! \cdot 3!)}}{{7! / (2! \cdot 5!)}}$, which reduces to $\frac{5! \cdot 2!}{3! \cdot 5!} = \frac{2!}{3!} = \frac{2}{6} = \frac{1}{3}$. However, the correct simplification should be $\frac{35}{21} = \frac{5}{3}$.

Question 14

Suppose a traveler wanted to visit a museum, an art gallery, and the state capitol building. 45 -minute tours are offered at each attraction hourly from 10 a.m. through 3 p.m. (6 different hours). Solve the problem, disregarding travel time .
-In how many ways could the traveler schedule all three tours in one day, with the art gallery being The last tour of the day?

Accepted Answer

The answer of Suppose a traveler wanted to visit a...

Question 15

Solve the problem.
-A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 7 different flags Available, how many possible signals can be flown?

Accepted Answer

Since there are 7 different flags available and 3 positions to fill where order matters and no flag can be used more than once, the number of possible signals is calculated using permutations: 7P3 = 7! / (7-3)! = 7*6*5 = 210.

Question 16

Solve the problem.
-A license plate is to consist of 2 letters followed by 3 digits. Determine the number of different License plates possible if repetition of letters and numbers is not permitted.

Accepted Answer

The answer of Solve the problem.
-A license plate is to...

Question 17

Solve the problem.
-How many ways can the letters in the word ʺWISCONSINʺ be arranged?

Accepted Answer

The word "WISCONSIN" has 9 letters with the following repetitions: I (2 times), S (2 times), and N (2 times). The formula for permutations of a set of objects where some objects are indistinguishable is n! / (p1! * p2! * ... * pk!), where n is the total number of objects, and p1, p2, ..., pk are the numbers of objects that are identical. Here, n = 9, and we have three sets of identical letters (I, S, N), each set having 2 identical letters. Thus, the number of arrangements = 9! / (2! * 2! * 2!) = 362,880 / (2 * 2 * 2) = 362,880 / 8 = 45,360.

Question 18

Evaluate the expression.
-$\frac { 9 ^ { C _ { 3 } } } { { } _ { 9 } P _ { 3 } }$

Accepted Answer

The expression simplifies to $\frac{9C3}{9P3}$. $9C3$ calculates the number of combinations of 9 items taken 3 at a time, while $9P3$ calculates the number of permutations of 9 items taken 3 at a time. The formula for combinations is $nCk = \frac{n!}{k!(n-k)!}$ and for permutations is $nPk = \frac{n!}{(n-k)!}$. Plugging in the values, we get $9C3 = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!}$ and $9P3 = \frac{9!}{(9-3)!} = \frac{9!}{6!}$. Simplifying the expression $\frac{9C3}{9P3}$ gives $\frac{\frac{9!}{3!6!}}{\frac{9!}{6!}} = \frac{1}{3!} = \frac{1}{6}$, since $3! = 6$.

Question 19

Evaluate the expression.
-$${ } _ { 9 } \mathrm { C } _ { 9 }$$

Accepted Answer

The expression ${ } _ { 9 } \mathrm { C } _ { 9 }$ represents the number of ways to choose 9 items out of 9, which is always 1, as there is exactly one way to choose all items.

Question 20

Evaluate the expression.
-$\frac { { } _ { 7 } ^ { \mathrm { P } _ { 4 } } } { { } _ { 10 } \mathrm { C } _ { 4 } }$

Accepted Answer

Permutation of 7 objects taken 4 at a time is equal to combination of 10 objects taken 4 at a time.

Quiz 11: Probability

Solve the problem. -A license plate is to consist of 3 letters followed by 5 digits. Determine the number of different License plates possible if the first letter must be an N , M , or P and repetition of letters and Numbers is not permitted.

Evaluate the expression. - ${ } _ { 8 } C _ { 0 }$

Solve the problem. -There are 9 horses in a race. In how many ways can the first three positions of the order of the Finish occur? (Assume there are no ties.)

Evaluate the expression. - ${ } _ { 7 } \mathrm { C } _ { 4 }$

Solve the problem. -In how many ways can the letters in the word PAYMENT be arranged if the letters are taken 4 at a Time?

Solve the problem. -How many ways can a president, vice-president, secretary, and treasurer be chosen from a club With 9 members? Assume that no member can hold more than one office.

Solve the problem. -How many different three-digit numbers can be written using digits from the set {2, 3, 4, 5, 6} Without any repeating digits?

Solve the problem. -How many different three-number ʺcombinationsʺ are possible on a combination lock having 22 Numbers on its dial? Assume that no numbers repeat. (Combination locks are really permutation Locks.)

Solve the problem. -A license plate is to consist of 2 letters followed by 4 digits. Determine the number of different License plates possible if repetition of letters and numbers is permitted.

Evaluate the expression. - $\frac { { } _ { 7 } C _ { 4 } } { { } _ { 7 } C _ { 2 } }$

Solve the problem. -A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 7 different flags Available, how many possible signals can be flown?

Solve the problem. -A license plate is to consist of 2 letters followed by 3 digits. Determine the number of different License plates possible if repetition of letters and numbers is not permitted.

Solve the problem. -How many ways can the letters in the word ʺWISCONSINʺ be arranged?

Evaluate the expression. - $\frac { 9 ^ { C _ { 3 } } } { { } _ { 9 } P _ { 3 } }$

Evaluate the expression. - ${ } _ { 9 } \mathrm { C } _ { 9 }$

Evaluate the expression. - $\frac { { } _ { 7 } ^ { \mathrm { P } _ { 4 } } } { { } _ { 10 } \mathrm { C } _ { 4 } }$