Question

evaluate $_{8}mathrm{p}_{4}$ and $_{8}mathrm{c}_{1}$.
(if necessary, consult a list of formulas.)
$_{8}mathrm{p}_{4}=\square$
$_{8}mathrm{c}_{1}=\square$

Explanation:

Step1: Recall permutation formula

The formula for permutation \( _nP_r \) is \( \frac{n!}{(n - r)!} \). For \( _8P_4 \), \( n = 8 \) and \( r = 4 \).
\[
_8P_4=\frac{8!}{(8 - 4)!}=\frac{8!}{4!}
\]
Since \( 8! = 8\times7\times6\times5\times4! \), we can cancel out \( 4! \):
\[
_8P_4 = 8\times7\times6\times5 = 1680
\]

Step2: Recall combination formula

The formula for combination \( _nC_r \) is \( \frac{n!}{r!(n - r)!} \). For \( _8C_1 \), \( n = 8 \) and \( r = 1 \).
\[
_8C_1=\frac{8!}{1!(8 - 1)!}=\frac{8!}{1!7!}
\]
Since \( 8! = 8\times7! \) and \( 1! = 1 \), we can cancel out \( 7! \):
\[
_8C_1=\frac{8\times7!}{1\times7!}=8
\]

Answer:

\( _8P_4 = 1680 \)
\( _8C_1 = 8 \)