Question

Number of $7$ digit number the sum of whose digits $61$ is :

• A. $12$
• B. $24$
• C. $28$
• D. none

Answer 1

Answer: (C)

$28$

Let $"x_1{x}_2{x}_3{x}_4{x}_5{x}_6{x}_7"$ denotes a seven digit number

$\therefore x_1+x_2+x_3+x_4+x_5+x_6+x_7=61$

Now possible ways:- $5$ nines and sum of remaining two $=16$

$\therefore x_m+x_n=16$

Possible ways:- $8+8$

$9+7$

$\therefore$ Number of ways with nines and eights $=\frac{7!}{5!2!}=21$

$\therefore$ Number of ways with nines and seven$=\frac{7!}{6!}=7$

$\therefore$ Total number of ways $=21+7$ $=28$.