(xi) $C_0^2-C_1^2+C_2^2-C_3^2+...=\begin{Bmatrix} 0, & \text { if } n \text { is odd } \\ (-1)^\frac{n}{2} \frac{^nC_n}{2},& \text { if } n \text { is even } \end{Bmatrix}$

(xii) $C_0C_1+C_1C_2+...+C_{n-1}C_n=^{2n}C_{n-1}$

(xiii) $C_0C_r+C_1C_{r+1}+...+C_{n-r}C_n=(^{2n}C_{n-r}) \text { or } (^{2n}C_{n+r})$

Greatest Term:

In the expansion of $(x+a)^n$

(i) If $\dfrac{n+1}{\frac{x}{a}+1}$ is an integer $= P (\text { Say})$ , then the greatest term is $T_p=T_p+1$ .

(ii) If $\dfrac{n+1}{\frac{x}{a}+1}$ is not an integer with $m$ as integral part of $\dfrac{n+1}{\frac{x}{a}+1}$ , then $T_{m+1}$ is the greatest term.

Greatest Coefficient:

In the expansion of $(x+a)^n$

(i) If $n$ is even, then the greatest coefficient is $\dfrac{^nC_n}{2}$ .

(ii) If $n$ is odd, then the greatest coefficient is $\dfrac{^nC_{n-1}}{2}$ or $\dfrac{^nC_{n+1}}{2}$ both being equal.

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