The Binomial Theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. The theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this formula: - \(\sum\) denotes summation.
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For a nonnegative integer and commuting quantities ,In expanding the factors, obtain by selecting the positions that contribute . There are such choices. For example,The coefficients are row five of Pascal's triangle, in reverse order under the displayed convention; the row is symmetric. Setting gives , the number of subsets of a five-element set. Each member has two inclusion choices, agreeing with the rule of product and the cardinality of the power set.