Binomial Theorem — JEE Main · 2010 · 11 Apr 2010 · Offline Question with Solution

Let

S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}

S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}

and

S_{3} = \sum_{j = 1}^{10} j^{2}^{10} C_{j}

.

Statement-1:

S_{3} = 55 \times 2^{9}

Statement-2:

S_{1} = 90 \times 2^{8}

and

S_{2} = 10 \times 2^{8}

(A)Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
(B)Statement-1 is true, Statement-2 is false✓ Correct
(C)Statement-1 is false, Statement-2 is true
(D)Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Answer: (B)

Solution

Using

j (j 10) = 10 (j - 1 9)

and

j (j - 1) (j 10) = 90 (j - 2 8)

, the sums collapse:

S_{1} = 90 \sum (j - 2 8) = 90 \cdot 2^{8}

and

S_{2} = 10 \sum (j - 1 9) = 10 \cdot 2^{9}

. Writing

j^{2} = j (j - 1) + j

gives

S_{3} = S_{1} + S_{2} = 90 \cdot 2^{8} + 10 \cdot 2^{9} = 90 \cdot 2^{8} + 20 \cdot 2^{8} = 110 \cdot 2^{8} = 55 \cdot 2^{9}

, so Statement-1 is correct. Statement-2 is wrong where it claims

S_{2} = 10 \cdot 2^{8}

; in fact

S_{2} = 10 \cdot 2^{9}

. Hence Statement-1 is true but Statement-2 is false.

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option