Exercise 401Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

The binomial theorem

#binomial theorem#expansion#coefficients

Statement

Prove that $(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$ .

Hint

Count the number of ways to choose $k$ factors of $b$ out of $n$ .

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Solution

Combinatorial proof. Expand $(a+b)^n=(a+b)(a+b)\cdots(a+b)$ ( $n$ factors). Each term in the expansion is a product of $n$ choices, one from each factor, giving $a^{n-k}b^k$ for each choice of $k$ positions where we pick $b$ . There are $\binom{n}{k}$ such choices.

Conclusion. $(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$ .

The binomial theorem

Statement

Solution

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