Question

Find the coefficient of x^7 in the expansion of (3x+4)^11

Answer 1

Use the binomial theorem:

$(a+b)^n=\displaystyle\sum_{k=0}^n\binom nka^{n-k}b^k$

In this case,

$(3x+4)^{11}=\displaystyle\sum_{k=0}^{11}\binom{11}k(3x)^{11-k}4^k$

The $x^7$ term occurs for $k=4$:

$\dbinom{11}4(3x)^{11-4}4^4=\dfrac{11!}{4!(11-4)!}3^74^4x^7=184,757,760x^7$