Final answer:
The sum ₂₀∑ r² ²⁰Cr is associated with the binomial theorem and represents a second moment about the mean for the binomial distribution. It can be calculated using properties of binomial coefficients, and the answer is 380 x 219.
Step-by-step explanation:
The student is asking about a sum involving binomial coefficients from the expansion of a binomial raised to a power using the binomial theorem. Specifically, they are interested in the sum ₂₀∑ r² ²⁰Cr where r varies from 0 to 20. We can recognize this as using the second moment about the mean for the binomial distribution.
The binomial theorem states that:
(a + b)n = an + nan-1b + … + n(n-1)/2! · an-2b2 + …
Here we have a=1 and b=x for the given question.
The coefficient ²⁰Cr represents the number of ways to choose r elements out of 20, and is the coefficient of xr in the expansion. The sum ₂₀∑ r² ²⁰Cr can be solved using properties of binomial coefficients and is equal to 380 x 219, which matches option C.