Question

evaluate the expression. express your answer in exact simplest form.

8c5·3c2=□

Explanation:

Step1: Calculate \(_{8}C_{5}\)

Use the combination formula \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\).
For \(_{8}C_{5}\), \(n = 8\), \(r=5\).
\(_{8}C_{5}=\frac{8!}{5!(8 - 5)!}=\frac{8!}{5!3!}=\frac{8\times7\times6\times5!}{5!\times3\times2\times1}=56\).

Step2: Calculate \(_{3}C_{2}\)

For \(_{3}C_{2}\), \(n = 3\), \(r = 2\).
\(_{3}C_{2}=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!\times1}=3\).

Step3: Multiply the two results

\(_{8}C_{5}\times_{3}C_{2}=56\times3 = 168\).

Answer:

\(168\)