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. $42$
• B. $82$
• C. $77$
• D. $35$

Answer 1

Answer: (C)

$77$

Coefficient of $x^{10}$ in $(x+x^2+x^3{)}^7=$ coefficient of $x^3$ in $(1+x+x^2{)}^7=$ coefficient of $x^3$ in $(1-x^3{)}^7(1-x{)}^{-7}$

${=}^{7+3-1}{C}_3-7$

${=}^9{C}_3-7$

$=\frac{9\times 8\times 7}{6}-7=77$

Alternate:

The digits are $1,1,1,1,1,2,3$ or $1,1,1,1,2,2,2$

Hence number of seven digit numbers formed $=\frac{7!}{5!}+\frac{7!}{4!3!}=77$