The value $\sum_{\mathrm{r}=0}^{22}{ }^{22} \mathrm{C}_{\mathrm{r}}{ }^{23} \mathrm{C}_{\mathrm{r}}$ is

(A) ${ }^{45} \mathrm{C}_{23}$

(B) ${ }^{44} \mathrm{C}_{23}$

(C) ${ }^{45} \mathrm{C}_{24}$

(D) ${ }^{44} \mathrm{C}_{22}$

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Binomial Theorem

The value

\sum_{r = 0}^{22}^{22} C_{r}^{23} C_{r}

Answer: (A)

Replace

(r 23)

(23 - r 23)

, so the sum is

r = 0 \sum 22 (r 22) (23 - r 23)

, the number of ways to pick

23

objects from a block of

22

and a block of

23

. By Vandermonde's identity

r \sum (r m) (k - r n) = (k m + n)

with

m = 22, n = 23, k = 23

, this equals

(23 45) .

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option