If \(\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}\), find x.

Question

Evaluate \(\frac{n!}{\left(n-r\right)!}\), when
(i) n = 6, r = 2 (ii) n = 9, r = 5

Answer 1

Given, \(\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}\)

\(⇒\frac{1}{6!}+\frac{1}{7\times6!}=\frac{x}{8\times7\times6!}\)

\(⇒\frac{1}{6!}(1+\frac{1}{7})=\frac{x}{8\times7\times6!}\)

\(⇒1+\frac{1}{7}=\frac{x}{8\times7}\)

\(⇒\frac{8}{7}=\frac{x}{8\times7}\)

\(⇒x=\frac{8\times8\times7}{7}\)

\(∴x=64\)

Solution and Explanation