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

Step-by-step solution

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 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 × coefficient of x in 1 - x - 8 = 4 + 8 - 1 C 8 - 1 - 8 × 1 + 8 - 1 C 8 - 1 = 11 C 7 - 8 × 8 C 7 = 11 × 10 × 9 3 × 8 4 × 3 × 2 - 8 × 8 = 330 - 64 = 26 6