alleryBy the Binomial Theorem, the expansion of $(a+b)^n$ for positive integer $n$ is
A$\sum_{r=0}^{n} \binom{n}{r}\,a^{n-r}\,b^r$
B$\sum_{r=0}^{n} \binom{n}{r}\,a^r\,b^{n-r}$ (both A and B are equivalent)
C$\binom{n}{0}\,a^n + \binom{n}{n}\,b^n$
D$a^n + n\,a\,b + b^n$
Answer & Solution
Correct answer: A. $\sum_{r=0}^{n} \binom{n}{r}\,a^{n-r}\,b^r$
1. syllabus states the Binomial Theorem: $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r}\,a^{n-r}\,b^r$.
2. Note: option B's formula $\sum \binom{n}{r}\,a^r\,b^{n-r}$ is also correct because $\binom{n}{r} = \binom{n}{n-r}$ (symmetric). However, A is the standard syllabus form.
3. Coefficients $\binom{n}{r}$ are called BINOMIAL COEFFICIENTS, and they appear in PASCAL'S triangle row $n$.
4. The expansion has $n+1$ terms (one for each value $r = 0, 1, \ldots, n$).
5. Options C, D are missing most of the terms — only first and last, or a wrong partial form.
_Source: syllabus Mathematics, Ch 7 "Binomial Theorem", §7.2 (Statement of the theorem), p. 1–3._
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