The-number-of-integers-which-lie-between-1-and-10-6-and-which-have-sum-of-digits-equal-to-12-is-

Question Number 21800 by Tinkutara last updated on 04/Oct/17

The number of integers which lie

between 1 and 10^{6} and which have sum

of digits equal to 12 is

Commented by mrW1 last updated on 27/Dec/17

For example to find such numbers

with 6 digits :

let abcdef be such a number . we have

a + b + c + d + e + f = 12 \dots (i)

1 ⩽ a ⩽ 9

0 ⩽ b, c, d, e, f ⩽ 9

number of solutions for (i) is the

coefficient of x^{12} in

(x + x^{2} + \dots + x^{9}) {(1 + x + x^{2} + \dots + x^{9})}^{5}

= x (1 + x + x^{2} + \dots + x^{8}) {(1 + x + x^{2} + \dots + x^{9})}^{5}

= \frac{x (1 - x^{9}) {(1 - x^{10})}^{5}}{{(1 - x)}^{6}}

let [x^{n}] denote the coefficient of term x^{n}

[x^{12}] \frac{x (1 - x^{9}) {(1 - x^{10})}^{5}}{{(1 - x)}^{6}}

= [x^{11}] \frac{(1 - x^{9}) {(1 - x^{10})}^{5}}{{(1 - x)}^{6}}

= [x^{11}] (1 - x^{9} - {5 x}^{10}) \underset{k = 0}{\sum^{\infty}} C_{k}^{k + 5} x^{k}

= C_{11}^{16} - C_{2}^{7} - {5 C}_{1}^{6} = 4317

similarily

5 digits : C_{11}^{15} - C_{2}^{6} - {4 C}_{1}^{5} = 1330

4 digits : C_{11}^{14} - C_{2}^{5} - {3 C}_{1}^{5} = 342

3 digits : C_{11}^{13} - C_{2}^{4} - {2 C}_{1}^{3} = 66

2 digits : C_{11}^{12} - C_{2}^{3} - C_{1}^{2} = 7

\Rightarrow Σ = 6062

Commented by Tinkutara last updated on 09/Oct/17

Thank you very much Sir!