Question

The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1$, $2$ and $3$ only is:

• A. $77$
• B. $42$
• C. $35$
• D. $82$

Answer 1

The correct option is A

$77$

Explanation for correct option

Case $\left(i\right)$: The digits are $1,1,1,1,1,2,3$

Therefore number of ways$=\frac{7!}{5!}=42$[$5!$ since $1$ is repeated $5$times]

Case $\left(ii\right)$: The digits are $1,1,1,1,2,2,2$

Therefore number of ways$=\frac{7!}{3!4!}=35$[$4!$ since $1$ is repeated $4$times and $3!$ since $2$ is repeated $3$ times]

Total number of ways$=35+42=77$

Hence, the correct option is $\left(\mathrm{A}\right)$