Question

Consider seven digit number $x_1,x_2,.......x_7$ , where $x_1,x_2,.......x_7,\ne 0$ having the property that $x_4$ is the greatest digit and all digits towards the left and right of $x_4$ are in decreasing order. Then total number of such number in which all digits are distinct is $a$, then find the sum of digits of $a$.

• A. $120$
• B. $360$
• C. $720$
• D. none of these

Answer 1

The correct option is C $720$

Number of selections of $7$ digits out of the digit $1,2,3,.....,9{=}^9{C}_7$

Number of digits out of these $7$ selected digits excluding the greatest digit $=6$

These $6$ digits can be divided in two group each having $3$ digits

$=\frac{6!}{3!3!2!}{=}^6{C}_3\times \frac{1}{2!}$

But the $3$ digits on one side can go on the other side.

$\therefore$Required number of ways

${=}^9{C}_7{.}^6{C}_3\times \frac{2}{2!}$

${=}^9{C}_2{.}^6{C}_3$

$=\frac{9!}{7!2!}\times \frac{6!}{3!3!}$

$=\frac{9\times 8}{2}\times \frac{6\times 5\times 4}{3\times 2}$

$=9\times 4\times 5\times 4=720$