${C_{0}}^{2} + {C_{1}}^{2} + {C_{2}}^{2} + {C_{3}}^{2} + . . . . . . . ?$

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Solution

$\begin{matrix} C_{0}^{2} + C_{1}^{2} + C_{3}^{3} + . . . . . . . . ? {(1 + n)}^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . . . . +^{n} C_{n} x^{n} {(x + 1)}^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} +^{n} C_{x} x^{n - 2} + . . . +^{n} C_{n} {(1 + x)}^{2 n} = [^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . . . . +^{n} C_{n} x^{n}] \times [^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} +^{n} C_{x} x^{n - 2} + . . . +^{n} C_{n}] C o e f f i c i e n t o f x^{n} :^{2 n} C_{n} = n C_{0}^{2} + n C_{1}^{2} + n C_{3}^{3} + . . . . . . . . +^{n} C_{n} . \end{matrix}$

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Similar questions

n = 11

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + . . . . + {(- 1)}^{n} C_{n}^{2}

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . + C_{n} x^{2}$ , then

$C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + . . . . . . . . . . + C_{n}^{2}$ =

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

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + . . . . . . . + C_{n}^{2}

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

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + . . . (- 1)^{n} C_{n}^{2}

n = 15

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

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