What is the coefficient of the x⁶y³ term in the expansion of (x + 2y)⁹? 
84, 168, 336, 672

Solution:

Given: Expression is (x + 2y)⁹

General term of a binomial expansion (a + b)ⁿ is T_{r + 1} = ⁿC_r a^{n - r}b^r

For (x + 2y)⁹, putting n =9, a = x, b = 2y

T_{r + 1} = ⁹C_r (x)^{9 - r}.(2y)^r

T_{r + 1} = ⁹C_r (x)^{9 - r}.(y)^r.(2)^r

Comparing with x⁶y³, we get

⇒ r = 3

T_{r + 1} = ⁹C₃ (x)^{9 - 3}.(y)³.(2)³

= ⁹C₃(2)³ × x⁶ × y³ 

= 84 × 8 × x⁶y³

= 672 x⁶y³

Hence, the coefficient of x⁶y³ is 672.

What is the coefficient of the x⁶y³ term in the expansion of (x + 2y)⁹?

Summary:

The coefficient of the x⁶y³ term in the expansion of (x + 2y)⁹ is 672.