Question

The coefficient of xⁿyⁿ in the expansion of {(1 + x) (1 + y) (x + y)}ⁿ is –

• A. $\sum_{r = 0}^{n}C_{r}^{2}$
• B. $\sum_{r = 0}^{n}C_{r}^{3}$
• C. $\sum_{r + s = 0}^{n}{nC_{r}^{n}C_{s}^{2}}$
• D. None

Answer 1

Answer: (B)

$\sum_{r = 0}^{n}C_{r}^{3}$

Answer 2

We have,

{(1 + x) (1 + y) (x + y)}ⁿ

= (C₀ + C₁x + C₂x² + .... + C_nxⁿ) ×

(C₀yⁿ + C₁y^n–1 + ... + C_n–1 y + C_n) ×

(C₀xⁿ + C₁ x^n–1y + ….. + C_n–1 x y^n–1 + C_n yⁿ)

Clearly, (Coefficient of xⁿ yⁿ on RHS)

= $C_{0}^{3} + C_{1}^{3} + C_{2}^{3} + .... + C_{n}^{3} = \sum_{r = 0}^{n}C_{r}^{3}$

Hence (2) is correct answer.