Question

The number of eight-digit integers, with the sum of digits equal to $12$ and formed by using the digits $1, \, 2$ and $3$ only are

• A. $255$
• B. $277$
• C. $288$
• D. $266$

Answer 1

Answer: (D)

The desired number is the coefficient of $x^{12}$ in $\left(x + x^{2} + x^{3}\right)^{8}$
= coefficient of $x^{12}$ in $x^{8}\left(1 + x + x^{2}\right)^{8}$
= coefficient of $x^{4}$ in $\left(\frac{1 + x^{3}}{1 - x}\right)^{8}$
= coefficient of $x^{4}$ in $\left(1 + x^{3}\right)^{8}\left(1 - x\right)^{- 8}$
= coefficient of $x^{4}$ in $\left(1 - 8 x^{3} + . . . .\right)\left(1 - x\right)^{- 8}$
= coefficient of $x^{4}$ in $\left(1 - x\right)^{- 8}-8\times $ coefficient of $x$ in $\left(1 - x\right)^{- 8}$
$=^{4 + 8 - 1}C_{8 - 1}-8\times ^{1 + 8 - 1}C_{8 - 1}$
$=^{11}C_{7}-8\times ^{8}C_{7}=\frac{11 \times 10 \times \cancel{9} 3 \times \cancel{8}}{\cancel{4} \times \cancel{3} \times \cancel{2}}-8\times 8$
$=330-64=266$