Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Properties of Binomial Co-efficient

(i) ^nC_r=^nC_{n-r}(r < n)

(ii) ^nC_r=^nC_y\Rightarrow (x=y)\text { or } (x+y=n)

(iii) ^nC_0+^nC_1+^nC_2+...+^nC_n=2^n

(iv)  ^nC_0-^nC_1+^nC_2-^nC_3+...+(-1)^n ^nC_n=0

(v) (1+x)^{-1}=1-x+x^2-x^3+...+(-1)^r x^r+...

(vi)  (1+x)^{-2}=1-2x+3x^2-...+(-1)^r (r+1)x^r+...

(vii) (1-x)^{-1}=1+x+x^2+x^3+...+x^r+...

(viii) (1-x)^{-2}=1+2x+3x^2+...+(r+1)x^r+...

(ix) C_0+C_2+C_4+...=C_1+C_3+C_5+...=2^{n-1}

(x) C_0^2+C_1^2+C_2^2+...+C^2_n=\dfrac{(2n)!}{(n!)^2}=^{2n}C_n

(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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