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

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

Tardigrade Admin · Accepted Answer

The desired number is the coefficient of

x^{12}

in

(x + x^{2} + x^{3})^{8}

= coefficient of

x^{12}

in

x^{8} (1 + x + x^{2})^{8}

= coefficient of

x^{4}

in

(\frac{1 + x ^{3}}{1 - x})^{8}

= coefficient of

x^{4}

in

(1 + x^{3})^{8} (1 - x)^{- 8}

= coefficient of

x^{4}

in

(1 - 8 x^{3} + ....) (1 - x)^{- 8}

= coefficient of

x^{4}

in

(1 - x)^{- 8} - 8 \times

coefficient of

x

in

(1 - x)^{- 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 9 3 \times 8}{4 \times 3 \times 2} - 8 \times 8

= 330 - 64 = 266

Q. 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

$255$

$277$

$288$

$266$

Q. 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

255

277

288

266

Q. 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

$255$

$277$

$288$

$266$