Question 1

Solve the problem.
-Suppose there are 8 roads connecting town A to town B and 6 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

Accepted Answer

To travel from A to C via B, a person can choose any of the 8 roads from A to B and then any of the 6 roads from B to C. Therefore, the total number of ways is 8 * 6 = 48 ways.

Question 2

Evaluate the factorial expression.
-$\frac { 7 ! } { 5 ! 2 ! }$

Accepted Answer

The expression simplifies to $\frac{7 \times 6}{2 \times 1} = 21$ because $5!$ cancels out in the numerator and denominator, leaving $7 \times 6$ in the numerator and $2!$ in the denominator.

Question 3

Solve the problem.
-Six strangers arrive at a business seminar and each person shakes hands with every other person. How many handshakes are there?

Accepted Answer

This problem can be solved using the formula for combinations since each handshake involves a unique pair of people. 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 at a time. Here, n = 6 (the six strangers) and k = 2 (since a handshake involves 2 people). Thus, the number of handshakes is C(6, 2) = 6! / [2!(6 - 2)!] = (6*5) / (2*1) = 15.

Question 4

Evaluate the factorial expression.
-$\frac { 6 ! } { 4 ! }$

Accepted Answer

The expression $\frac{6!}{4!}$ simplifies to $\frac{6 \times 5 \times 4!}{4!}$, where the $4!$ terms cancel out, leaving $6 \times 5 = 30$.

Question 5

Evaluate the factorial expression.
-$\frac { 4 ! } { 6 ! }$

Accepted Answer

The factorial expression $\frac { 4 ! } { 6 ! }$ simplifies to $\frac { 1 } { 5 \times 6 } = \frac { 1 } { 30 }$ because $6! = 6 \times 5 \times 4!$ and the $4!$ terms cancel out.

Question 6

Evaluate the factorial expression.
-$\frac { 7 ! } { ( 7 - 4 ) ! }$

Accepted Answer

The answer of Evaluate the factorial expression.
-\(\frac { 7 !...

Question 7

Evaluate the factorial expression.
-15 !

Accepted Answer

The factorial of 15 (15!) is the product of all positive integers up to 15, which equals 1,307,674,368,000 or $1.3077 \times 10^{12}$.

Question 8

Solve the problem.
-A computer printer allows for optional settings with a panel of four on-off switches in a row. How many different settings can be selected if no three adjacent switches can all be off?

Accepted Answer

To solve this, consider the total number of settings without restriction: 2^4 = 16. Now, subtract the cases where three adjacent switches are off. There are 3 such cases: (000X, 0000, X000). Thus, 16 - 3 = 13 valid settings.

Question 9

Solve the problem.
-A sports shop sold tennis rackets in 3 different weights, 3 types of string, and 4 grip sizes. How many different rackets could be sold?

Accepted Answer

The total number of different rackets is calculated by multiplying the number of options for each feature: 3 weights * 3 types of string * 4 grip sizes = 36 different rackets.

Question 10

Solve the problem.
-How many odd three-digit numbers can be written using digits from the set $\{ 2,3,4,5,6 \}$ if no digit may be used more than once?

Accepted Answer

There are 5 choices for the hundreds digit, 4 choices for the tens digit, and 3 choices for the units digit. So, there are $5 \times 4 \times 3 = 60$ possible three-digit numbers. However, half of these numbers are even, so there are $60 \div 2 = 30$ odd three-digit numbers.

Question 11

Evaluate the factorial expression.
-$\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }$, where $\mathrm { n } = 38$ and $\mathrm { r } = 9$

Accepted Answer

This expression is the formula for combinations, given by $C(n, r) = \frac{n!}{r!(n-r)!}$. It calculates the number of ways to choose $r$ elements out of $n$ without regard to the order. Substituting $n = 38$ and $r = 9$, we get $C(38, 9)$, which equals $163,011,640$.

Question 12

Solve the problem.
-A salesperson packed 3 shirts and 5 ties. With one shirt, he could wear all 5 ties. With another shirt, he could wear 4 ties. With the third shirt, he could wear only 2 ties. How many different Combinations did he have?

Accepted Answer

The total combinations are calculated by adding the combinations for each shirt: 5 (first shirt) + 4 (second shirt) + 2 (third shirt) = 11 combinations.

Question 13

Evaluate the factorial expression.
-$\frac { 8 ! } { 6 ! 2 ! }$

Accepted Answer

The expression can be simplified by canceling out the common factorials. $8! = 8 \times 7 \times 6!$ and $6!2! = 6! \times 2 \times 1$, so $\frac{8!}{6!2!} = \frac{8 \times 7 \times 6!}{6! \times 2 \times 1} = \frac{8 \times 7}{2 \times 1} = 28$.

Question 14

Solve the problem.
-A restaurant offered salads with 3 type(s)of dressing and 6 different topping(s). How many different types of salad could be ordered?

Accepted Answer

The answer of Solve the problem.
-A restaurant offered salads with...

Question 15

Evaluate the factorial expression.
-10!

Accepted Answer

The factorial of 10 (denoted as 10!) is the product of all positive integers from 1 to 10, which equals 3,628,800.

Question 16

Solve the problem.
-In how many ways can the number 44 be written as the sum of two primes?

Accepted Answer

44 can be written as the sum of two primes in 3 ways: 3 + 41, 11 + 33, and 17 + 27.

Question 17

A shirt company has 3 designs, each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

Accepted Answer

The total number of different types of shirts is calculated by multiplying the number of designs (3) by the number of sleeve options (2) and then by the number of color patterns (7): 3 * 2 * 7 = 42 types.

Question 18

Evaluate the factorial expression.
-$\frac { 7 ! } { 6 ! }$

Accepted Answer

The expression simplifies to 7 because $7! = 7 \times 6!$ and when divided by $6!$, the $6!$ terms cancel out, leaving just 7.

Question 19

Solve the problem.
-How many three-digit numbers have the sum of their digits equal to 20?

Accepted Answer

To find three-digit numbers whose digits sum to 20, we can use combinatorics, considering the problem as distributing 20 identical items into 3 distinct boxes (digits), where each box can hold between 1 and 9 items (since a digit can be 1-9, and the first digit cannot be 0 for a three-digit number). This is a variation of the stars and bars method, adjusted for the constraints. The direct calculation for this specific problem is complex due to the constraints on digit values, so a more practical approach involves enumerating possibilities or using programming. However, the correct enumeration, considering the constraints (especially that the first digit cannot be 0 and each digit must be between 0 and 9), gives us 36 valid combinations.

Question 20

Evaluate the factorial expression.
-$\frac { n ! } { ( n - r ) ! }$, where $n = 46$ and $r = 10$

Accepted Answer

The expression $\frac { n ! } { ( n - r ) ! }$ simplifies to $n(n-1)(n-2)...(n-r+1)$. For $n = 46$ and $r = 10$, it becomes $46 \times 45 \times 44 \times ... \times 37$. Calculating this product or using a calculator/computer, the result is approximately $1.479 \times 10^{16}$, which matches option A.

Quiz 10: Counting Methods

Solve the problem. -Suppose there are 8 roads connecting town A to town B and 6 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

Evaluate the factorial expression. - $\frac { 7 ! } { 5 ! 2 ! }$

Solve the problem. -Six strangers arrive at a business seminar and each person shakes hands with every other person. How many handshakes are there?

Evaluate the factorial expression. - $\frac { 6 ! } { 4 ! }$

Evaluate the factorial expression. - $\frac { 4 ! } { 6 ! }$

Evaluate the factorial expression. - $\frac { 7 ! } { ( 7 - 4 ) ! }$

Evaluate the factorial expression. -15 !

Solve the problem. -A computer printer allows for optional settings with a panel of four on-off switches in a row. How many different settings can be selected if no three adjacent switches can all be off?

Solve the problem. -A sports shop sold tennis rackets in 3 different weights, 3 types of string, and 4 grip sizes. How many different rackets could be sold?

Solve the problem. -How many odd three-digit numbers can be written using digits from the set $\{ 2,3,4,5,6 \}$ if no digit may be used more than once?

Evaluate the factorial expression. - $\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }$ , where $\mathrm { n } = 38$ and $\mathrm { r } = 9$

Solve the problem. -A salesperson packed 3 shirts and 5 ties. With one shirt, he could wear all 5 ties. With another shirt, he could wear 4 ties. With the third shirt, he could wear only 2 ties. How many different Combinations did he have?

Evaluate the factorial expression. - $\frac { 8 ! } { 6 ! 2 ! }$

Solve the problem. -A restaurant offered salads with 3 type(s) of dressing and 6 different topping(s) . How many different types of salad could be ordered?

Evaluate the factorial expression. -10!

Solve the problem. -In how many ways can the number 44 be written as the sum of two primes?

A shirt company has 3 designs, each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

Evaluate the factorial expression. - $\frac { 7 ! } { 6 ! }$

Solve the problem. -How many three-digit numbers have the sum of their digits equal to 20?

Evaluate the factorial expression. - $\frac { n ! } { ( n - r ) ! }$ , where $n = 46$ and $r = 10$

Quiz 10: Counting Methods

Solve the problem. -Suppose there are 8 roads connecting town A to town B and 6 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

Evaluate the factorial expression. - 7!5!2!\frac { 7 ! } { 5 ! 2 ! }5!2!7!​

Solve the problem. -Six strangers arrive at a business seminar and each person shakes hands with every other person. How many handshakes are there?

Evaluate the factorial expression. - 6!4!\frac { 6 ! } { 4 ! }4!6!​

Evaluate the factorial expression. - 4!6!\frac { 4 ! } { 6 ! }6!4!​

Evaluate the factorial expression. - 7!(7−4) !\frac { 7 ! } { ( 7 - 4 ) ! }(7−4) !7!​

Evaluate the factorial expression. -15 !

Solve the problem. -A computer printer allows for optional settings with a panel of four on-off switches in a row. How many different settings can be selected if no three adjacent switches can all be off?

Solve the problem. -A sports shop sold tennis rackets in 3 different weights, 3 types of string, and 4 grip sizes. How many different rackets could be sold?

Solve the problem. -How many odd three-digit numbers can be written using digits from the set {2,3,4,5,6}\{ 2,3,4,5,6 \}{2,3,4,5,6} if no digit may be used more than once?

Evaluate the factorial expression. - n!r!(n−r) !\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }r!(n−r) !n!​ , where n=38\mathrm { n } = 38n=38 and r=9\mathrm { r } = 9r=9

Solve the problem. -A salesperson packed 3 shirts and 5 ties. With one shirt, he could wear all 5 ties. With another shirt, he could wear 4 ties. With the third shirt, he could wear only 2 ties. How many different Combinations did he have?

Evaluate the factorial expression. - 8!6!2!\frac { 8 ! } { 6 ! 2 ! }6!2!8!​

Solve the problem. -A restaurant offered salads with 3 type(s) of dressing and 6 different topping(s) . How many different types of salad could be ordered?

Evaluate the factorial expression. -10!

Solve the problem. -In how many ways can the number 44 be written as the sum of two primes?

A shirt company has 3 designs, each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

Evaluate the factorial expression. - 7!6!\frac { 7 ! } { 6 ! }6!7!​

Solve the problem. -How many three-digit numbers have the sum of their digits equal to 20?

Evaluate the factorial expression. - n!(n−r) !\frac { n ! } { ( n - r ) ! }(n−r) !n!​ , where n=46n = 46n=46 and r=10r = 10r=10

Evaluate the factorial expression. - $\frac { 7 ! } { 5 ! 2 ! }$

Evaluate the factorial expression. - $\frac { 6 ! } { 4 ! }$

Evaluate the factorial expression. - $\frac { 4 ! } { 6 ! }$

Evaluate the factorial expression. - $\frac { 7 ! } { ( 7 - 4 ) ! }$

Solve the problem. -How many odd three-digit numbers can be written using digits from the set $\{ 2,3,4,5,6 \}$ if no digit may be used more than once?

Evaluate the factorial expression. - $\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }$ , where $\mathrm { n } = 38$ and $\mathrm { r } = 9$

Evaluate the factorial expression. - $\frac { 8 ! } { 6 ! 2 ! }$

Evaluate the factorial expression. - $\frac { 7 ! } { 6 ! }$

Evaluate the factorial expression. - $\frac { n ! } { ( n - r ) ! }$ , where $n = 46$ and $r = 10$