binomial theorem - Determine the coefficient of $x^6$ in $(1 - x)^{15}$

I was asked to determine the coefficient of $x^6$ for $(1-x)^{15}$.

I used the binomial theorem as follows:

$$^{15}C_0 (1)^{15}(-x)^0 + \, ^{15}C_1 (1)^{14} (-x)^1 + \cdots + \, ^{15}C_6 (1)^9(-x)^6$$

then I evaluated $^{15}C_6$ and got $5005$. So would the coefficient simply be $5005$? Also, lets say the exponent for the $-x$ term was odd would I have made the overall coefficient $-5005$?

Sorry just trying to understand the basics of expanding the binomial theorem. Any help would be greatly appreciated.

Answer

Yes, the answer is $5005$ and if the exponent were odd the coefficient would be negative for exactly the reason you say.