Quick answer

C(n, k) = n! / (k! (n − k)!). Equivalently, multiply k descending factors from n and divide by k!.

Formula

  • C(n, k) = (n)(n − 1)…(n − k + 1) / k!
  • Symmetry: C(n, k) = C(n, n − k)

Introduction

If you already know what a combination counts, the formula is the machinery. If notation still feels fuzzy, read what a binomial coefficient is first, then return here for algebra.

Hand calculation rewards partial products: write the rising numerator, write k!, cancel shared factors, then divide. Jumping straight to 20! on a calculator often triggers overflow even when the final answer is modest.

After you simplify by hand, plug the same n and k into the binomial coefficient examples walkthroughs to see how textbook stories map to numbers.

What the factorial fraction represents

Imagine lining up n distinct items and choosing k positions for a "selected" label. There are n! ways to permute all items. Fixing which k are selected removes k! reorderings inside the chosen block and (n − k)! reorderings inside the unchosen block, leaving n! / (k! (n − k)!) distinct label patterns.

The product form is the same idea without writing full factorials. Multiply k factors starting at n and descending: n, n − 1, …, n − k + 1. That product counts ordered picks if you stopped after k steps. Dividing by k! collapses each group of k! orderings into one combination.

Step-by-step breakdown

  • Step 1: Confirm 0 ≤ k ≤ n.
  • Step 2: If k > n/2, replace k with n − k (symmetry).
  • Step 3: Build numerator (n)(n−1)…(n−k+1).
  • Step 4: Divide by k!.

Example: C(9, 4). Symmetry is optional here. Numerator = 9×8×7×6 = 3024. Denominator = 4×3×2×1 = 24. Quotient = 126.

Cancellation trick: write 8×7×6×5 / (4×3×2×1) and strike 4×2 from top and bottom before multiplying. Exams reward this discipline because it reduces arithmetic errors.

Step-by-step guide

  1. Choose factorial or product form. Product form is faster on paper for k ≤ 6. Factorial form is compact in proofs.
  2. Apply symmetry when k is large. C(100, 98) should be computed as C(100, 2) = (100×99)/2 = 4950.
  3. Cancel before multiplying huge chains. Look for factors that appear in both numerator and denominator.
  4. Sanity-check sign and size. Combinations are nonnegative integers. If you obtain a fraction, revisit cancellation.

Two routes to C(11, 4)

Product route: (11×10×9×8) / (4×3×2×1) = 7920 / 24 = 330.

Factorial route: 11! / (4! 7!) = (11×10×9×8) / 24 after canceling 7! from numerator and denominator. Same answer, less clutter when you cancel early.

Spreadsheet users can type =COMBIN(11,4) for audits, but exams still expect the manual structure above.