JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Binomial Theorem

If the sum of the coefficients in the expansion of

(x + y)^{n}

is 4096 , then the greatest coefficient in the expansion is

Answer: 924

Solution

The sum of coefficients in

(x + y)^{n}

is obtained by setting

x = y = 1

(1 + 1)^{n} = 2^{n}

.

Given:

2^{n} = 4096 = 2^{12}

, so

n = 12

.

The coefficients in the expansion are

(0 12), (1 12), \dots, (12 12)

.

The greatest coefficient occurs at the middle term(s). For

n = 12

(even), the maximum is at

r = \frac{n}{2} = 6

(6 12) = \frac{12 !}{6 ! \cdot 6 !} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{665280}{720} = 924

JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Integer answer