For r - 0-1-2- n- Co Cr - C1Cr-1 - C2Cr-2 -Cn-r Cn - 2nCn-r and hence C2o - C21 - C22 - - - C2a - 2nCn

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Combination

For r = 0,1,2...

Question

For r = 0,1,2,.............................. n,
$C_{o} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} . . . . . . . . . . . . C_{n - r} C_{n} = 2 n C_{(n + r)}$ and hence $C_{o}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . . . . . . . + C_{a}^{2} = {2 n}_{n}^{C}$

True

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r = 0, 1, 2, . . . ., n

C_{0} \cdot C_{r} + C_{1} \cdot C_{r + 1} + C_{2} \cdot C_{r + 2} + . . . . + C_{n - r} \cdot C_{n} =^{2 n} C_{(n + r)}

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . + C_{n}^{2} =^{2 n} C_{n}

C_{0} \cdot C_{1} + C_{1} \cdot C_{2} + C_{2} \cdot C_{3} + . . . . . + C_{n - 1} \cdot C_{n} =^{2 n} C_{n + 1}

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}

C_{0} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} + . . . C_{n - r} C_{n} =

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{n} x^{n}

C_{0} C_{r} + C_{1} C_{r + 1} + . . . . . . + C_{n - r} C_{n} = \frac{(2 n)!}{(n + r)! (n - r)!}

Find the value of $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} . . . . . C_{n}^{2}$ , where $C_{r} =^{n} C_{r}$

^{2 n} C_{n} = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + \dots \dots + C_{n}^{2}

^{2 n} C_{n} =

x

(1 + x)^{n} {(1 + \frac{1}{x})}^{n}

^{2 n} C_{n} = \frac{1.3.5.7 \dots \dots (2 n - 1)}{n!}

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