Question

Use the formula for nCr to evaluate the expression. 10c3/6c4

Answer 1

SOLUTION

The given expression is

$\frac{^{10}C_3}{^6C_4}$

Recall the formula for combination

$^nC_r=\frac{n!}{(n-r)!r!}$

Solving the given expresssion

$\frac{^{10}C_3}{^6C_4}=\frac{10!}{(10-3)!3!}\div\frac{6!}{(6-4)!4!}$

This further gives

$\begin{gathered} \frac{10!}{(10-3)!3!}\div\frac{6!}{(6-4)!4!} \\ =\frac{10!}{7!3!}\div\frac{6!}{2!4!} \\ =\frac{10!}{7!3!}\times\frac{2!4!}{6!} \end{gathered}$

This further gives

$\begin{gathered} =\frac{10\times9\times8\times7!}{7!\times3\times2!}\times\frac{2!4!}{6\times5\times4!} \\ =\frac{10\times9\times8}{6\times5\times3} \\ =8 \end{gathered}$

Therefore the solution is 8