Question

Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number. The numbers of such numbers if
$x_1<x_2<x_3\le x_4<x_5<x_6$ is

• A. ${}^9{C}_4$
• B. ${}^9{C}_3$
• C. ${}^{10}{C}_4$
• D. ${}^{10}{C}_3$

Answer 1

The correct option is C ${}^{10}{C}_4$

When $x_1<x_2<x_3<x_4<x_5<x_6$
$($Since we have to form $6$ digit number, $\therefore x_1\ne 0)$
Numbers can be selected in${}^9{C}_6$ ways and can be arranged in $1$ way.

When $x_1<x_2<x_3=x_4<x_5<x_6$
Numbers can be selected in${}^9{C}_5$ ways and can be arranged in $1$ way.

$\therefore$ required ways
$={}^9{C}_6+{}^9{C}_5={}^{10}{C}_6={}^{10}{C}_4$
$[\because {}^n{C}_{r-1}+{}^n{C}_r={}^{n+1}{C}_r]$