If the sum of the coefficients in the expansion of (x + y)ⁿ is 4096, then the greatest coefficient in the expansion is _____________.

Solution

Maths sir · 3 steps

Put x = y = 1 to get 2ⁿ = 4096, so n = 12, then pick the middle binomial coefficient.

1. Find n. Setting x = y = 1 makes every term equal to its coefficient, so the sum of the coefficients is 2ⁿ. Since 2ⁿ = 4096 = 2¹², we get n = 12.
2. Where it peaks. The coefficients C(12, r) rise up to the middle and then fall again. For even n there is exactly one middle term, at r = n/2 = 6. That is the 7th term.

   C(12, 6) = (12 × 11 × 10 × 9 × 8 × 7)/(6 × 5 × 4 × 3 × 2 × 1) = 665280/720 = 924

3. So. The greatest coefficient in the expansion of (x+y)¹² is 924.

The common slip: 792. Taking the 6th term as the middle term uses C(12, 5) = 792. The middle term is the 7th, C(12, 6).

The idea to keep. Sum of coefficients means put every variable equal to 1. The greatest binomial coefficient is C(n, n/2) for even n, and C(n, (n-1)/2) = C(n, (n+1)/2) for odd n.

Answer from the NTA final answer key. Solution by PrepWiser.