Question

evaluate each expression. sample problem ( _8p_3 ) (\frac{8!}{(8-3)!} = \frac{8!}{5!} = 336) ( _8c_4 ) > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Recall the combination formula

The formula for combinations is \( _nC_r=\frac{n!}{r!(n - r)!} \). For \( _8C_4 \), we have \( n = 8 \) and \( r=4 \).

Step2: Substitute into the formula

Substitute \( n = 8 \) and \( r = 4 \) into the formula: \( _8C_4=\frac{8!}{4!(8 - 4)!}=\frac{8!}{4!×4!} \).

Step3: Expand the factorials

We know that \( n!=n\times(n - 1)\times\cdots\times1 \), so \( 8! = 8\times7\times6\times5\times4! \) and \( 4! = 4\times3\times2\times1 \). Substituting these in, we get:

$$
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Step4: Calculate the numerator and denominator

Calculate the numerator: \( 8\times7\times6\times5=1680 \).
Calculate the denominator: \( 4\times3\times2\times1 = 24 \).

Step5: Divide the numerator by the denominator

\( \frac{1680}{24}=70 \).

Answer:

70