Question

Evaluate the following expression Express your answer as a fraction or a decimal number rounded to four decimal places13P8 ÷ 13C7

Answer 1

Answer:

30,240

Explanation:

We were given the expression:

$\frac{13P_8}{13C_7}$

We will simplify the expression as shown below:

$\begin{gathered} \frac{13P_{8}}{13C_{7}} \\ \text{For Permutation, we have:} \\ nP_r=\frac{n!{}}{(n-r)!} \\ 13P_8=\frac{{13!}}{(13-8)!}=\frac{13!}{5!} \\ 13P_8=\frac{13\times12\times11\times10\times9\times8\times7\times6\times5!}{5!} \\ 13P_8=13\times12\times11\times10\times9\times8\times7\times6 \\ 13P_8=51,891,840 \\ \\ \begin{equation*} 13C_7 \end{equation*} \\ \text{For Combination, we have:} \\ nC_r=\frac{n!}{r!(n-r)!} \\ 13C_7=\frac{13!}{7!(13-7)!}=\frac{13!}{7!6!} \\ 13C_7=\frac{13\times12\times11\times10\times9\times8\times7!}{7!\times6\times5\times4\times3\times2\times1} \\ 13C_7=\frac{13\times12\times11\times10\times9\times8}{6\times5\times4\times3\times2\times1} \\ 13C_7=\frac{1,235,520}{720} \\ 13C_7=1,716 \\ \\ \Rightarrow\frac{13P_{8}}{13C_{7}}=\frac{51,891,840}{1,716}=30,240 \\ \frac{13P_8}{13C_7}=30,240 \\ \\ \therefore\frac{13P_{8}}{13C_{7}}=30,240 \end{gathered}$