Question

1. - / 1 points evaluate the permutation. \\( _7p_4 \\) 3. - / 1 points evaluate the permutation. \\( p(8,8) \\) 4. - / 1 points evaluate the combination. \\( _7c_4 \\) 5. - / 1 points evaluate the combination. \\( \dbinom{8}{6} \\)

Explanation:

Problem 2: Evaluate \( _7P_4 \)

Step1: Recall permutation formula

The formula for permutations is \( _nP_r=\frac{n!}{(n - r)!} \), where \( n = 7 \) and \( r = 4 \).

Step2: Substitute values into formula

Substitute \( n = 7 \) and \( r = 4 \) into the formula: \( _7P_4=\frac{7!}{(7 - 4)!}=\frac{7!}{3!} \)

Step3: Expand factorials

We know that \( n!=n\times(n - 1)\times\cdots\times1 \), so \( 7! = 7\times6\times5\times4\times3! \) and \( 3! = 3\times2\times1 \). Then \( \frac{7!}{3!}=\frac{7\times6\times5\times4\times3!}{3!} \)

Step4: Simplify the expression

Cancel out \( 3! \) in the numerator and denominator: \( 7\times6\times5\times4 = 840 \)

Step1: Recall permutation formula

The formula for permutations is \( P(n,r)=\frac{n!}{(n - r)!} \), where \( n = 8 \) and \( r = 8 \).

Step2: Substitute values into formula

Substitute \( n = 8 \) and \( r = 8 \) into the formula: \( P(8,8)=\frac{8!}{(8 - 8)!}=\frac{8!}{0!} \)

Step3: Recall the value of \( 0! \)

We know that \( 0! = 1 \), and \( 8! = 8\times7\times6\times5\times4\times3\times2\times1 \)

Step4: Calculate the value

\( \frac{8!}{0!}=\frac{40320}{1}=40320 \)

Step1: Recall combination formula

The formula for combinations is \( _nC_r=\frac{n!}{r!(n - r)!} \), where \( n = 7 \) and \( r = 4 \).

Step2: Substitute values into formula

Substitute \( n = 7 \) and \( r = 4 \) into the formula: \( _7C_4=\frac{7!}{4!(7 - 4)!}=\frac{7!}{4!3!} \)

Step3: Expand factorials

\( 7! = 7\times6\times5\times4! \), \( 4! = 4\times3\times2\times1 \), \( 3! = 3\times2\times1 \). So \( \frac{7!}{4!3!}=\frac{7\times6\times5\times4!}{4!\times3\times2\times1} \)

Step4: Simplify the expression

Cancel out \( 4! \) in the numerator and denominator: \( \frac{7\times6\times5}{3\times2\times1}=\frac{210}{6} = 35 \)

Answer:

\( 840 \)

Problem 3: Evaluate \( P(8,8) \)