Question

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

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

Answer 1

The correct option is C $28$

Let $"x_1,x_2,x_3,x_4,x_5,x_6,x_7"$ denote 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 numbers of ways $=21+7$

$=28$