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:

Finding the number of ways formed by the seven-digit number by using the digits $1,2$ and $3$ that sum is equal to $10$ .

Let us consider,

We can divide the number $10$ in two ways by using the digits $1,2$ and $3$ there are,

Case I: $\left\{1,1,1,1,1,2,3\right\}$

Case II: $\left\{2,2,2,1,1,1,1\right\}$

Thus,

Case I: The number of ways a seven-digit number can be formed:

The number of places to be filled is equal to $7$and the number of repetitions is equal to $5$, this can be expressed as

$$\begin{gathered} \Rightarrow \frac{7!}{5!} \\ \Rightarrow \frac{7\times 6\times 5!}{5!} \\ \Rightarrow 42 \end{gathered}$$

Case II: The number of ways a seven-digit number can be formed:

The number of places to be filled is equal to $7$and the number of repetitions is equal to $3$ and $4$, for the numbers $2\mathrm{and}1$, respectively, this can be expressed as

$$\begin{gathered} \Rightarrow \frac{7!}{3!\times 4!} \\ \Rightarrow \frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!} \\ \Rightarrow 35 \end{gathered}$$

The total number of ways is,

$\Rightarrow 42+35=77$

Therefore, the correct answer is Option C.