Combinations Summary

Combinatorics · Axiom Academy

Let's review the key concepts of unordered selection from Unit 3. Definition: A combination is an unordered selection of r objects from n distinct objects Order Doesn't Matter: A, B, C is the same as C, A, B - only the items selected matter, not their arrangement Notation: Written as C(n,r), nCr , or most commonly n choose r Key Difference: Unlike permutations where ABC ≠ BAC, combinations treat these as identical selections Formula: C(n,r) = n! / (r! × (n-r)!) Why Divide by r!: We start with n!/(n-r)! permutations, then divide by r! to eliminate duplicate orderings Relationship to Permutations: C(n,r) = P(n,r) / r! because each combination corresponds to r! different permutations Simplification Tip: Cancel factorials before computing: C(10,8) = C(10,2) = (10 × 9)/(2 × 1) = 45 Symmetry: C(n,r) = C(n,n-r) - choosing r items to include equals choosing n-r items to exclude Pascal's Identity: C(n,r) = C(n-1,r-1) + C(n-1,r) - the foundation of Pascal's Triangle Boundary Conditions: C(n,0) = 1 and C(n,n) = 1 - exactly one way to choose nothing or everything Pascal's Triangle: Each entry is the sum of the two entries above it, displaying all combination values Definition: Selecting r items from n types where items can be repeated (order still doesn't matter) Formula: C(n+r-1, r) = C(n+r-1, n-1) - surprisingly larger than regular combinations! Why It Works: Equivalent to distributing r identical objects into n distinct bins

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