Question

For r = 0, 1,...., let Ar, Br and Cr denote, respectively, the coefficient of Xr in the expansions of (1 + x)30, (1 + x)10 and (1 + x)20. Then Σr=110 Ar (B10 Br − C10 Ar) is equal to

• A. (B10 − C10)
• B. A10 (B102 − C10 A10)
• C. 0
• D. C10 − B10

Answer 1

Answer: (D)

C10 − B10

Answer 2

Step 1. Express coefficients.
Ar = C(30,r), Br = C(10,r), Cr = C(20,r).

Step 2. Split the sum.
Σ Ar (B10 Br − C10 Ar)
= B10 Σ Ar Br − C10 Σ Ar2.

Step 3. Recognize binomial convolution.
Σr=110 C(30,r)·C(10,r) = [coefficient of x20 in (1+x)30] − 1 = C(30,20) − 1.
Σr=110 [C(30,r)]2 = [coefficient of x10 in (1+x)20] − 1 = C(20,10) − 1.

Step 4. Substitute and simplify.
B10=C(10,10), C10=C(20,10).
Result = C(10,10)[C(30,20)−1] − C(20,10)[C(20,10)−1]
= C(20,10) − C(10,10) = C10 − B10.