JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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

Choose the correct answer:

Explanation

Let $S = \sum_{r=0}^{22} ({}^{22}C_{r})({}^{23}C_{r})$

Consider, $(1 + x)^{22} = {}^{22}C_{0}x^{0} + {}^{22}C_{1}x^{1} + \dots + {}^{22}C_{22}x^{22} \dots (1)$ Again, $(x + 1)^{23} = {}^{23}C_{0}x^{23} + {}^{23}C_{1}x^{22} + \dots + {}^{23}C_{23}x^{0} \dots (2)$

$(1) \times (2)$ gives

$(1 + x)^{45} = ({}^{22}C_{0}x^{0} + {}^{22}C_{1}x^{1} + \dots + {}^{22}C_{22}x^{22}) \times ({}^{23}C_{0}x^{23} + {}^{23}C_{1}x^{22} + \dots + {}^{23}C_{23}x^{0}) \dots (3)$

Again, $S = \sum_{r=0}^{22} ({}^{22}C_{22-r})({}^{23}C_{r})$ ,

Using ${}^{n}C_{r} = {}^{n}C_{n-r}$ $\Rightarrow S = ({}^{22}C_{22})({}^{23}C_{0}) + ({}^{22}C_{21})({}^{23}C_{1}) + \dots + ({}^{22}C_{0})({}^{23}C_{22})$

From $(3)$ , comparing coefficient of $x^{23}$ on both sides

or $S = {}^{45}C_{23}$