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The number of terms in the expansion of ...

The number of terms in the expansion of `(a+b+c)^n` where `ninN` is

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`(a+b+c)^(n) = [a+(b+c)]^(n)`
`=a^(n)+.^(n)C_(1)a^(n-1)(b+c)^(1) + .^(n)C_(2)a^(n-2)(b+c)^(2)+"….."+.^(n)C_(n) (b+c)^(n)`
Further, expanding each term of the R.H.S., we note that
First term consists of 1 term
Second term on simplification gives 2 terms
Third term on expansion gives 3 term
Fourth term on expansion gives 4 terms and so on.
`:.` Total number of terms `= 1 + 2 + 3 + "........" + (n+1)`
`= ((n+1)(n+2))/(2)`
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