Question

5. Determine the expansion of (2 + x)6 using the binomial theorem.

Answer 1

1 + 64x + 240x^2 + 480x^3 + 480x^4 + 192x^5 + x^6.

Explanation:

(2 + x)^6 = C(6, 0) * 2^6 * x^0 + C(6, 1) * 2^5 * x^1 + C(6, 2) * 2^4 * x^2 + C(6, 3) * 2^3 * x^3 + C(6, 4) * 2^2 * x^4 + C(6, 5) * 2^1 * x^5 + C(6, 6) * 2^0 * x^6.

C(6, 0) = 6! / (0! * (6-0)!) = 1,

C(6, 1) = 6! / (1! * (6-1)!) = 6,

C(6, 2) = 6! / (2! * (6-2)!) = 15,

C(6, 3) = 6! / (3! * (6-3)!) = 20,

C(6, 4) = 6! / (4! * (6-4)!) = 15,

C(6, 5) = 6! / (5! * (6-5)!) = 6,

C(6, 6) = 6! / (6! * (6-6)!) = 1

(2 + x)^6 = 1 * 2^6 * x^0 + 6 * 2^5 * x^1 + 15 * 2^4 * x^2 + 20 * 2^3 * x^3 + 15 * 2^2 * x^4 + 6 * 2^1 * x^5 + 1 * 2^0 * x^6.