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Question:

For r=0, 1, ..., 10, let Ar, Br and Cr denote, respectively, the coefficient of xr in the expansions of (1+x)10, (1+x)20 and (1+x)30. Then ∑r=110 Ar(B10Br - C10Ar) is equal to

0

C10 - B10

B10 - C10

A10(B20 - C10A10)

Solution:

The given expression can be written as ∑r=110 10Cr(20C10 20Cr - 30C10 10Cr)
20C10 ∑r=110 10Cr 20Cr - 30C10 ∑r=110 (10Cr)2
⇒ 20C10[30C20] - 30C10[20C10]
⇒ 30C10 - 30C10 ⇒ C10 - B10