Question

If a, b, c are positive integers such that a + b + c £ 8 , then the number of possible values of the ordered triplets (a, b, c) is

• A. 84
• B. 56
• C. 83
• D. 54

Answer 1

Answer: (B)

56

Answer 2

Here a + b + c = n, where n = 3, 4, 5, 6, 7, 8 because a, b, c are all positive integers.

The number of positive integral solutions of (a + b + c = n)

= Coefficient of xⁿ in (x + x³ …….)³ = ^n–1C₂

\ The required number of solutions =

$\sum_{n = 3}^{8}{n - 1C_{2}}$= ²C₂ + ³C₂+ ⁴C₂ + ⁵C₂ +⁶C₂ +⁷C₂

= ³C₀ + ³C₁ + ⁴C₂ + ⁵C₃ + ⁶C₄ + ⁷C₅
(Q ⁿC_r = ⁿC_n–r)

= ⁴C₁ + ⁴C₂ + ⁵C₃ + ⁶C₄ +⁷C₅ = ……..= ⁸C₅ = 56