Binomial Theorem Summary

Combinatorics · Axiom Academy

Let's review the key concepts, formulas, and applications from Unit 4. Statement: For any positive integer n , the expansion of ( x + y ) n is the sum of terms of the form C(n, k) x n-k y k Meaning: The theorem provides a formula to expand any binomial raised to a power without tedious multiplication Index Range: The index k runs from 0 to n , giving exactly n + 1 terms in the expansion Symmetry: Coefficients are symmetric because C( n, k ) = C( n, n-k ) Why Binomial Coefficients Are Combinations Counting Argument: The coefficient of x n-k y k counts the number of ways to choose k positions for y from n binomials Distributive Law: When expanding ( x + y ) n , we select either x or y from each of n factors Selection Process: To get y k , we choose k factors to contribute y and the rest contribute x Formula Connection: This selection is precisely what C( n, k ) = n !/( k !( n-k )!) counts Construction Rule: Each entry is the sum of the two entries directly above it, with edges always equal to 1 Row Meaning: Row n contains the binomial coefficients C( n , 0), C( n , 1), ..., C( n, n ) Recurrence: The construction reflects the identity C( n, k ) = C( n-1, k-1 ) + C( n-1, k ) Applications: Provides quick access to coefficients and reveals patterns in binomial expansions Sum of Row: C( n , 0) + C( n , 1) + ... + C( n, n ) = 2 n (set x = y = 1) Alternating Sum: C( n , 0) - C( n , 1) + C( n , 2) - ... = 0 (set x = 1, y = -1)

This is the written version of the interactive lesson above. See the full Combinatorics course.