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

Answer: (C)

Solution: Let $x_i(1\le i\le 7)$ be the digit at the $i$ th place. As $1\le x_i\le 3$, and $x_1+x_2+\dots +x_7=10$, at most one $x_i$ can be 3 .
Two cases arise.
Case 1.
Exactly one of $x_i$ 's is 3 . In this case exactly one of the remaining $x_i$ 's is 2 .
In this case, the number of seven digit numbers is
$\frac{7!}{5!}=7\times 6=42$
Case 2.
None of $x_i$ 's is 3 .
In this case exactly three of $x_i$ 's is 2 and the remaining four $x_i$ 's are 1.
In this case, the number of seven digit number is
$\frac{7!}{3!4!}=35$
Hence, the required seven digit numbers is $42+35=77$.