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Question
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
Options
`((n + 1)(n + 2))/2`
n + 1
n + 2
(n + 1)n
Solution
The number of terms in the expansion of (a + b + c)n, where n ∈ N is `((n + 1)(n + 2))/2`.
Explanation:
We have (a + b + c)n = [a + (b + c)]n
= an + nC1 an – 1 (b + c)1 + nC2 an – 2 (b + c)2 + ... + nCn (b + c)n
Further, expanding each term of R.H.S., we note that
First term consist of 1 term.
Second term on simplification gives 2 terms.
Third term on expansion gives 3 terms.
Similarly, fourth term on expansion gives 4 terms and so on.
The total number of terms = 1 + 2 + 3 + ... + (n + 1)
= `((n + 1)(n + 2))/2`
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