The value of sum_r=0^22 ^22C_r \ ^23C_r is — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)

The value of

\sum_{r=0}^{22} {}^{22}C_r \; {}^{23}C_r

A.
${}^{44}C_{23}$
B.
${}^{44}C_{22}$
C.
${}^{45}C_{23}$
D.
${}^{45}C_{24}$

Show answer and solution

Correct option: C

Solution

Using the Vandermonde-Chu identity: ∑(r=0 to n) C(m,r)·C(n,r) = C(m+n, n).

Here, ∑(r=0 to 22) C(22,r)·C(23,r) = ∑(r=0 to 22) C(22,r)·C(23,23-r) [since C(23,r) = C(23,23-r)].

This equals C(22+23, 23) = C(45,23) by Vandermonde's identity (choosing 23 items from 45 by splitting into two groups of 22 and 23).

Hence the answer is ⁴⁵C₂₃.

[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]

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