Quick answer
Translate words to n and k, then evaluate C(n, k).
Formula
- Committee: C(n,k)
- Lottery: C(n,k) before probability weights
Introduction
Read the story twice: once for what is being counted, once for whether order matters. Most wrong answers come from the second read, not arithmetic.
When formulas feel abstract, revisit what a binomial coefficient means with the committee picture fresh in mind.
Probability exercises often stop at the count before multiplying p terms; our binomial coefficients in probability article shows where C(n, k) sits inside the binomial PMF.
How to read each example template
Setup states the population size n and selection size k. Work shows either the product form or a symmetry shortcut. Result states C(n, k) in standard notation.
Listing small cases remains underrated. For C(4, 2), write all six pairs once in your notebook; later problems feel less magical.
Quick reference
- C(n,k) = n!/(k!(n−k)!)
- Symmetry: C(n,k)=C(n,n−k)
Keep a scratch column for symmetry checks. If k is more than half of n, switch to n − k before you multiply long chains.
Step-by-step guide
- Highlight clue phrases. "Choose," "select," "committee," and "without regard to order" point to combinations.
- Assign n and k. Distinguish pool size from selection size; do not swap them when the sentence mentions both.
- Compute and label. Write C(n, k) = value so graders see structure.
Four worked stories
Committee: 12 candidates, choose 4. C(12, 4) = 495 distinct panels.
Lottery: pick 6 from 49. C(49, 6) = 13,983,816 lines (counting equally likely tickets before probability commentary).
Cards: five-card hand from 52. C(52, 5) = 2,598,960 hands.
Algebra: coefficient of x^3 in (1 + x)^7 is C(7, 3) = 35 because you choose three factors of x.
