"The value of $\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}$ is"

Question

Accepted Answer

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$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}\,.\,{}^{23}{C_r}}$

$= \sum\limits_{r = 0}^{22} {{}^{22}{C_r}\,{}^{23}{C_{23 - r}}}$ [using ${}^n{C_r} = {}^n{C_{n - r}}$]

$$ = {}^{22}{C_0}{}^{23}{C_{23}} + {}^{22}{C_1}{}^{23}{C_{22}}\, + \,...\, + \,{}^{22}{C_{21}}{}^{23}{C_2} + {}^{22}{C_{22}}{}^{23}{C_1}$$

We know,

$${(1 + x)^{22}} = {}^{22}{C_0} + {}^{22}{C_1}x + {}^{22}{C_2}{x^2} + {}^{22}{C_3}{x^3}\, + \,...\, + \,{}^{22}{C_{22}}{x^{22}}$$

and

$${(1 + x)^{23}} = {}^{23}{C_0} + {}^{23}{C_1}x + {}^{23}{C_2}{x^2}\, + \,...\, + \,{}^{23}{C_{22}}{x^{22}} + {}^{23}{C_{23}}{x^{23}}$$

Now coefficient of ${x^{23}}$ in ${(1 + x)^{22}}{(1 + x)^{23}}$ or ${(1 + x)^{45}}$

$$ = {}^{22}{C_0}{}^{23}{C_{23}} + {}^{22}{C_1}{}^{23}{C_{22}}\, + \,...\, + \,{}^{22}{C_{21}}\,.\,{}^{23}{C_2} + {}^{22}{C_{22}}\,.\,{}^{23}{C_1}$$

$= \sum\limits_{r = 0}^{22} {{}^{22}{C_r}\,.\,{}^{23}{C_{23 - r}}}$

$\therefore$ Coefficient of ${x^{23}}$ in ${(1 + x)^{45}} = {}^{45}{C_{23}}$

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The value of $\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}$ is