Question

The number of integers which lie between 1 and 10⁶ and which have the sum of the digits equal to 12 is-

• A. 8550
• B. 5382
• C. 6062
• D. 8055

Answer 1

Answer: (C)

6062

Answer 2

Consider the product (x⁰ + x¹ + x² +….+ x⁹) (x⁰ + x¹ + x² + ….. + x⁹) … 6 factors. The number of ways in which the sum of the digits will be equal to 12 is equal to the coefficient of x¹² in the above product. So, required number of ways

= coeff. of x¹² in (x⁰ + x¹ + x² + …. + x⁹)⁶

= coeff. of x¹² in $\left( \frac{1 - x^{10}}{1 - x} \right)^{6}$

= coeff. of x¹² in (1 – x¹⁰)⁶ (1 – x)^–6

= coeff. of x¹² in (1 – x)^–6 (1 – ⁶C₁ x¹⁰ +….)

= coeff. of x¹² in (1 – x)^–6 – ⁶C₁ . coeff. of x² in (1 – x)^–6

= ^{12 + 6 – 1}C_{6 – 1} – ⁶C₁ × ^{2 + 6 – 1}C_{6 – 1} = ¹⁷C₅ – 6 × ⁷C₅ = 6062.