Question

Evaluate _5C_3- O A. 10 O B. 20 O C. 60 O D. 30 O E. none of these

Answer 1

$\bf _nC_r=\cfrac{n!}{r!(n-r)!}\qquad \begin{cases} n=5\\ r=3 \end{cases}\implies _5C_3=\cfrac{5!}{3!(5-3)!} \\\\\\ _5C_3=\cfrac{5!}{3!(2)!}\implies _5C_3=\cfrac{5\cdot 4\cdot ~~\begin{matrix} 3\cdot 2\cdot 1 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{~~\begin{matrix} 3\cdot 2\cdot 1 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~(2\cdot 1)}\implies _5C_3=\cfrac{20}{2}\implies _5C_3=10$