Question

Determine the independent term of x^7 in the expansion of (3x^2 + 4)^12.

(a) 220  * 4^6

(b) 230

(c) 548* 3!

(d) 220 * 3^6 * 4^6

I got this question in class test.

The doubt is from Counting topic in chapter Counting of Discrete Mathematics

Answer 1

Correct choice is (d) 220 * 3^6 * 4^6

For explanation I would say: By using Binomial theorem = ^n∑k=0 (n^k) x^ky^n-k = n^0x^0y^n + n^1x^1y^n-1 + n^2x^2y^n-2 + … + n^nx^ny^0, where (n^k) = \(\frac{n!}{k!(n−k)!}\). Now, Tr+1 = ^nCra^n-rb^r, T9+1 = ^12C6a^12-6b^6 = 220 * (3x^2)^6 * (4)^6 = 220 * 3^6 * 4^6. Hence the coefficient is 220 * 3^6 * 4^6.