Question

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

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

Answer 1

The Correct Option is B

To solve the problem of finding the value of \(\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r\), we can use a combinatorial identity known as Vandermonde's identity. The identity states:

\(^{m+n}C_k = \sum_{r=0}^{k} {^mC_r} {^nC_{k-r}}\)

In this problem, we are given:

• \(m = 22\)
• \(n = 23\)
• \(k = 22 + 23 = 45\)

This means we can rewrite the given sum using Vandermonde's identity as:

\(\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r = ^{45}C_{22}\)

However, we need to match this with the correct form according to the options provided.

1. The options have \(^{45}C_{23}\). Notice that \(^{45}C_{22} = ^{45}C_{23}\) because of the symmetric property of combinations, where \(^nC_k = ^nC_{n-k}\).

Therefore, the equivalent form in the options is:

• Correct Answer: \(^{45}C_{23}\)

Hence, the value of the sum \(\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r\) is \(^{45}C_{23}\).