Question
The numbers of integers between 1 and 106 have the sum of their digit equal to K(where 0 < K < 18) is -
- (K + 6)C6 – (K – 4)C6
- KC6 – 6 KC4
- KC6 – 6 K – 4C6
- K + 6C6 – 6 K – 4C6
The correct answer is: K + 6C6 – 6 K – 4C6
The required no. of ways = no. of solution of the equation (x1 + x2 + x3 + x4 + x5 + x6 = K)
Where 0 xi 9, i = 1, 2, …6, where 0 < K < 18
= Coefficient of xK in (1 + x + x2 +….. + x9)6
= Coefficient of xK in 
= Coefficient of xk in (1 – 6x10 + 15 x20 – ….)
(1 + 6 C1x + 7 C2 x2 + …. +(7 – K – 10 – 1) CK–10 xK–10 + ….+(7 + K – 1) CK xK + …)
= k + 6CK – 6. K–4CK–10
= k + 6C6 – 6. K–4C6 .
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