Math miscellaneousHard
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 
Ar(B10Br - C10Ar) is equal to
Ar(B10Br - C10Ar) is equal toOptions
A.B10 - C10
B.A10(B102 - C10A10)
C.0
D.C10 - B10
Solution
Let y = Ar(B10Br - C10Ar)
ArBr = coefficient of x20 in ((1 + x)10 (x + 1)20) - 1
= C20 - 1 = C10 - 1 and(Ar)2 = coefficient of x10 in ((1 + x)10 (x + 1)10) - 1 = B10 - 1
⇒ y = B10(C10 - 1) - C10(B10 - 1) = C10 - B10.
Ar(B10Br - C10Ar)ArBr = coefficient of x20 in ((1 + x)10 (x + 1)20) - 1= C20 - 1 = C10 - 1 and
(Ar)2 = coefficient of x10 in ((1 + x)10 (x + 1)10) - 1 = B10 - 1⇒ y = B10(C10 - 1) - C10(B10 - 1) = C10 - B10.
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