Answer to Question #97692 in Algebra for belle

Question

Find the coefficient of X2 and x 3 in the expansion of (2-x)raised to the power of 6(

Answer 1

According to binomial expansion formula, $(x+y)^n = \sum_{k=0}^n C_k^n x^k y^{n-k}$, where $C_n^k = \frac{n!}{(n-k)!k!}$ is the binomial coefficient.

Hence, $(2-x)^6 = \sum_{k=0}^6 C_k^6 (-x)^k 2^{6-k}$ .

From the last formula, coefficients next to $x^2$, $x^3$ are $C_2^6 2^4 = 240$ and $C_3^6 (-1)^3 2^3 = -160$ respectively.