The value of $ \sum_\limits{r=0}^{22}{ }^{22} C_{r}{ }^{23} C_{r} $ is

JEE MATHEMATICS MCQ: BINOMIAL THEOREM

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The value of

\sum\limits_{{r=0}}^{22}{ }^{22} C_{r}{ }^{23} C_{r}

View answer and explanation

Correct option: D

{ }^{45} \mathrm{C}_{23}

Option D (

{ }^{45} \mathrm{C}_{23}

) is correct because the sum can be identified as the coefficient of

x^{23}

in the expansion of

(1+x)^{22}(1+x)^{23}

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\left({ }^{10} C_0\right)^2-\left({ }^{10} C_1\right)^2+\ldots-\left({ }^{10} C_9\right)^2+\left({ }^{10} C_{10}\right)^2

a

d

\left( {n + 1} \right)

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

(1-x)^{201}\left(1+x+x^2\right)^{200}

\left(\frac{1}{x}+2 x\right)^n

{x^5}

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

\left(\begin{array}{c}2 n+1 \\ 0\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 3\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 6\end{array}\right)+\ldots=170

n