Question

The numbers of integers between 1 and 10⁶ have the sum of their digit equal to K(where 0 < K < 18) is –

• A. ^{(K + 6)}C₆ – ^{(K – 4)}C₆
• B. ^KC₆ – 6 · ^KC₄
• C. ^KC₆ – 6 · ^{K – 4}C₆
• D. ^{K + 6}C₆ – 6 · ^{K – 4}C₆

Answer 1

Answer: (D)

^{K + 6}C₆ – 6 · ^{K – 4}C₆

Answer 2

The required no. of ways = no. of solution of the equation

(x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = K)

Where 0 £ x_i £ 9, i = 1, 2, …6, where 0 < K < 18

= Coefficient of x^K in (1 + x + x² +….. + x⁹)⁶

= Coefficient of x^K in $\left( \frac{1 - x^{10}}{1 - x} \right)^{6}$

= Coefficient of x^k in (1 – 6x¹⁰ + 15 x²⁰ – ….)

(1 + 6 C₁x + 7 C₂ x² + …. +(7 – K – 10 – 1)

C_K–10 x^K–10 + ….+(7 + K – 1) C_K x^K + …)

= ^{k + 6}C_K – 6. ^K–4C_K–10

= ^{k + 6}C₆ – 6. ^K–4C₆ .