This problem involves finding the value of r for which a specific sum of products of binomial coefficients is maximum. Let's break it down step by step.

Step 1: Understand the Expression
The given expression is: ${}_{20}C_r{}_{20}C_0+{}_{20}C_{r-1}{}_{20}C_1+{}_{20}C_{r-2}{}_{20}C_2+\dots +{}_{20}C_0{}_{20}C_r$ This can be written more compactly as: $S=\sum _{k=0}^r{}_{20}C_{r-k}{}_{20}C_k$

Step 2: Recognize the Combinatorial Identity
This sum is a standard combinatorial identity. It represents the number of ways to choose r items from two distinct groups of 20 items each. The identity is: $\sum _{k=0}^r{}_{n}C_{r-k}{}_{m}C_k={}_{n+m}C_r$ In our case, both n and m are 20. Therefore, the sum simplifies to: $S={}_{40}C_r$ So, the problem reduces to finding the value of r for which ${}_{40}C_r$ is maximum.

Step 3: Find the Maximum Binomial Coefficient
For a given n, the binomial coefficient ${}_{n}C_r$ is maximum when r is as close as possible to n/2. This is because the binomial distribution is symmetric and peaks at the center.
Here, n = 40. Since 40 is even, the maximum occurs at r = n/2 = 20.
However, we must check the options: 15, 10, 11, 20. The value 20 is an option.

Step 4: Verification
For even n, the binomial coefficient is maximum at r = n/2. So for n=40, r=20 gives the maximum value of ${}_{40}C_r$.

Final Answer
Therefore, the value of r for which the given expression is maximum is $20$.

Related Topics

Binomial Theorem and Coefficients: The binomial coefficient ${}_{n}C_r$ represents the number of ways to choose r items from n distinct items. The sequence of binomial coefficients for a fixed n is symmetric and unimodal, reaching its maximum at the center.

Formulae

Vandermonde's Identity: $\sum _{k=0}^r{}_{n}C_{r-k}{}_{m}C_k={}_{n+m}C_r$

Maximum Binomial Coefficient: For a given n, ${}_{n}C_r$ is maximum when: $r=⌊\frac{n}{2}⌋\text{ or }r=⌈\frac{n}{2}⌉$ (i.e., the integer(s) closest to n/2). If n is even, the maximum is unique at r = n/2.