The Binomial Theorem

Combinatorics · Axiom Academy

Discover the elegant formula for expanding powers of binomials For any non-negative integer n and any values x and y, we have: This means when we expand (x + y) n , we get a sum of n+1 terms. Each term has the form C(n,k) x n-k y k , where k ranges from 0 to n. C(n,k) = the binomial coefficient "n choose k" = n!/(k!(n-k)!) x n-k = the power of x decreases from n to 0 y k = the power of y increases from 0 to n When we expand (x + y) n , the sum runs from k = 0 to k = n. This gives us exactly n+1 terms. Notice how the powers of x decrease while the powers of y increase, and they always sum to n in each term. 3. Example: Expanding (x + y) 4 Let's apply the binomial theorem to expand (x + y) 4 : The coefficients (1, 4, 6, 4, 1) form the 4th row of Pascal's Triangle! 4. Example: Expanding (a + b) 5 Using the same pattern with n = 5: The binomial theorem has several useful special cases: 6. Why Does It Work? The Combinatorial View When we expand (x + y) n , we're multiplying n copies of (x + y) together: For each of the n factors, we must choose either x or y. To get a term with x n-k y k , we need to: Choose y from the remaining k factors The number of ways to select which k factors contribute a y is exactly C(n,k). This is why C(n,k) appears as the coefficient!

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