Question

1. - / 1 points evaluate the combination. c(8,8) 7. - / 1 points in how many ways can you select five people from a group of ten if the order of selection is important? ways 8. - / 1 points in how many ways can you select two people from a group of 20 if the order of selection is not important? ways 9. - / 1 points

Explanation:

Question 6

Step1: Recall combination formula

The formula for combinations is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n! = n\times(n - 1)\times\cdots\times1 \) and \( 0!=1 \). For \( C(8, 8) \), we have \( n = 8 \) and \( r = 8 \).

Step2: Substitute values into formula

Substitute \( n = 8 \) and \( r = 8 \) into the formula: \( C(8, 8)=\frac{8!}{8!(8 - 8)!}=\frac{8!}{8!0!} \). Since \( 0!=1 \) and \( \frac{8!}{8!}=1 \), we get \( \frac{1}{1}=1 \).

Step1: Identify permutation formula

When order is important, we use permutations. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 10 \) (total number of people) and \( r = 5 \) (number of people to select).

Step2: Substitute values into formula

Substitute \( n = 10 \) and \( r = 5 \) into the formula: \( P(10, 5)=\frac{10!}{(10 - 5)!}=\frac{10!}{5!} \). We know that \( 10! = 10\times9\times8\times7\times6\times5! \), so \( \frac{10\times9\times8\times7\times6\times5!}{5!}=10\times9\times8\times7\times6 \).

Step3: Calculate the product

Calculate \( 10\times9\times8\times7\times6 = 30240 \).

Step1: Recall combination formula

When order is not important, we use combinations. The formula for combinations is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n = 20 \) (total number of people) and \( r = 2 \) (number of people to select).

Step2: Substitute values into formula

Substitute \( n = 20 \) and \( r = 2 \) into the formula: \( C(20, 2)=\frac{20!}{2!(20 - 2)!}=\frac{20!}{2!18!} \). We know that \( 20! = 20\times19\times18! \), so \( \frac{20\times19\times18!}{2\times1\times18!} \).

Step3: Simplify the expression

Cancel out \( 18! \) from numerator and denominator, then calculate \( \frac{20\times19}{2\times1}=\frac{380}{2}=190 \).

Answer:

\( 1 \)

Question 7