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Question: The number of terms in the expansion of (a+b+c)<sup>n</sup>, where n∈N, is...

The number of terms in the expansion of (a+b+c)n, where n∈N, is

A

(n+1)(n+2)2\frac{(n + 1)(n + 2)}{2}

B

n+1

C

n+2

D

(n+1)n

Answer

(n+1)(n+2)2\frac{(n + 1)(n + 2)}{2}

Explanation

Solution

(a + (b+c))n = an +nC1 an−1(b + c)1+nC2an−2 (b + c)2 + + nCn (b + c)n.

Further expanding each term of R.H.S.,

First term on expansion gives one term.

Second term on expansion gives two terms.

Third term on expansion gives three terms and so on.

⇒ Total no. of terms = 1 + 2 + 3 + + (n + 1)

= (n+1)(n+2)2\frac{(n + 1)(n + 2)}{2}