Question

If a, b, c are odd positive integers, then number of integral solutions of a+b+c = 13 is

• A. 21
• B. 42
• C. 28
• D. 56

Answer 1

Answer: (A)

21

Answer 2

Let a = 2x + 1, b = 2y+1, c = 2z + 1 (where x, y, z, are non negative integral solution)

∴ (2x+1) + (2y+1) + (2z+1) = 13

⇒ x+y+z = 5

Required number of solutions

= Coefficient of a⁵ in (a⁰ + a¹ + a² + ...)³

= Coefficient of a⁵ in $\left( \frac{1}{1 - a} \right)^{3}$

= Coefficient of a⁵ in (1 – a)^-3

= $7C_{5} = \frac{7x6}{2} = 21$