Question

Consider seven digit number $x_1,x_2,.........,$ where $x_1,x_2,.......x_7,\ne 0$ having the property that 4 $x_4$ is greatest digit and the digits towards left and right are in decreasing order. Then total number of such numbers in which all digits are distinct is

• A. ${}^9{C}_7{\times }^6{C}_3\times 2$
• B. ${}^9{C}_6{\times }^5{C}_3$
• C. ${}^{10}{C}_7{\times }^6{C}_3$
• D. ${}^{10}{C}_3$

Answer 1

The correct option is A ${}^9{C}_7{\times }^6{C}_3\times 2$

From {1, 2, 3.....9}, select any 7 digit in ${}^9{C}_7$ ways.

Out of the selected digits place the highest digit in middle place.

From the remaining 6 digits select any three in ${}^6{C}_3$ ways and they can be arranged in two ways either on the left or right of middle digit.

$\therefore$ No. of ways ${=}^9{C}_7{.}^6{C}_3\times 2$