Math · College algebra · Concept
Why binomial coefficients appear
An expansion counts choices from repeated factors.
Choose one term from every factor
In (a+b)^n there are n identical factors. Each product term chooses either a or b from each factor. To obtain a^(n−k)b^k, choose exactly k factors to supply b.
Count choices without ordering them
The chosen positions form a subset, not an ordered list. There are n!/(k!(n−k)!) such subsets. This explains both the coefficient and the symmetry between k and n−k.
Pascal’s recurrence splits the possibilities
Separate subsets according to whether they include the last position. Those that include it choose k−1 from n−1; those that omit it choose k from n−1. Their counts add. This finite theorem assumes a nonnegative integer n.
Common mistakes
- Do not drop the mixed terms.
- In (a−b)^n, substitute −b and preserve its alternating powers.
Work through an example
Use coefficients, powers and signs together.
Expand (x−2)³ →Sources and scope
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