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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Binomial Theorem

If

(^{30} C_{1})^{2} + 2 (^{30} C_{2})^{2} + 3 (^{30} C_{3})^{2} + \dots + 30 (^{30} C_{30})^{2} = \frac{α \cdot 60 !}{( 30 ! ) ^{2}}

, then

α

is equal to

(A)30
(B)60
(C)15✓ Correct
(D)10

Answer: (C)

Solution

Let

S = \sum_{k = 1}^{30} k (k 30)^{2}

. Replacing

k

by

30 - k

and using

(k 30) = (30 - k 30)

gives the companion sum

S = \sum_{k = 0}^{30} (30 - k) (k 30)^{2}

. Adding the two,

2 S = 30 \sum_{k = 0}^{30} (k 30)^{2} = 30 (30 60)

by Vandermonde's identity. Thus

S = 15 (30 60) = 15 \cdot \frac{60 !}{( 30 ! ) ^{2}}

, so

α = 15

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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