Question

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

• A. $55$
• B. $66$
• C. $77$
• D. $88$

Answer 1

The correct option is C

$77$

Explanation for correct option:

Forming $7$digit number:

The digits with possible combinations to form $7$- digit number whose sum is $10$ are

Case $1$: $1,1,1,1,1,2,3=\frac{7!}{5!}$

Case $2$: $1,1,1,1,2,2,2=\frac{7!}{4!3!}$.

So total Number of possible seven-digit numbers is

$$\begin{gathered} =\frac{7!}{5!}+\frac{7!}{\left(4!\right)\left(3!\right)} \\ =\frac{7\times 6\times 5!}{5!}+\frac{7\times 6\times 5\times 4!}{3\times 2\times 4!} \\ =42+35 \\ =77 \end{gathered}$$

Therefore, the number of seven-digit integer formed is $77$.

Hence, the correct answer is option (C).