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The $(r+1)^\text{th}$ term (the GENERAL term) in the expansion of $(a+b)^n$ is
A$T_{r+1} = \binom{n}{r}\,a^{n-r}\,b^r$
B$T_{r+1} = \binom{n}{r}\,a^r\,b^{n-r}$
C$T_{r+1} = \binom{n}{r+1}\,a^{n-r}\,b^r$
D$T_{r+1} = n!\,a^{n-r}\,b^r$
Answer & Solution
Correct answer: A. $T_{r+1} = \binom{n}{r}\,a^{n-r}\,b^r$
1. NCERT §7.2.1 (General Term): writing out the binomial expansion term by term ($r = 0, 1, 2, \ldots$):
- $T_1$: $r = 0$, term = $\binom{n}{0}\,a^n\,b^0$
- $T_2$: $r = 1$, term = $\binom{n}{1}\,a^{n-1}\,b^1$
- $T_{r+1}$: $r$-th index, term = $\binom{n}{r}\,a^{n-r}\,b^r$.
2. So the GENERAL term is $T_{r+1} = \binom{n}{r}\,a^{n-r}\,b^r$.
3. Important: the index of $T$ is $r+1$ (not $r$) — the very first term corresponds to $r=0$.
4. Option B has the powers swapped (wrong unless $b$ is the leading term). Option C has wrong binomial coefficient. Option D omits the binomial coefficient.
_Source: NCERT Class 11 Mathematics, Ch 7, §7.2.1 (General Term, Eq. for $T_{r+1}$), p. 4._
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