If f(y)=1-(y-1)+(y-1)^2-(y-1)^3+-(y-1)^17, then the coefficient of y^2 in it is:

Question

If $f(y)=1-(y-1)+(y-1)^2-(y-1)^3+\cdots-(y-1)^{17}$, then the coefficient of $y^2$ in it is:

Accepted Answer

Coefficient of $y^2$ in $(y-1)^n$ is ${}^nC_2(-1)^{n-2}$. Summing over $n=2$ to $17$ with alternating signs (matching the given series structure):

$${}^2C_0+{}^3C_1+{}^4C_2+{}^5C_3+\cdots+{}^{17}C_{15}$$

Using the hockey-stick identity repeatedly, this telescopes to ${}^{18}C_{15}={}^{18}C_3$.

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