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Introduction
Simple Combinations help when we select items and the order does not matter. Use combinations when you care only about which items are chosen - not the sequence in which they appear.
This pattern is important because many real-world selection problems (forming teams, choosing committee members, selecting lottery numbers) require combinations rather than permutations.
Pattern: Simple Combinations (Order Doesn’t Matter)
If you need to choose r items from n distinct items and order is irrelevant, use nCr.
Formula:
nCr = n! / (r! (n - r)!)
Shortcut idea: Compute as nPr / r! (permutations divided by the r! orders of each selection).
Step-by-Step Example
Question
From 6 students, a team of 3 students is to be chosen. In how many ways can the team be selected?
Solution
-
Step 1: Identify what is given.
Total students = n = 6; team size = r = 3. Order does not matter. -
Step 2: Choose the combination formula.
UsenCr = n! / (r! (n - r)!). -
Step 3: Substitute and compute.
6C3 = 6! / (3! × 3!) = (6 × 5 × 4 × 3!)/(3! × 3!) = (6 × 5 × 4) / (3 × 2 × 1) = 120 / 6 = 20 -
Final Answer:
The team can be selected in 20 different ways. -
Quick Check:
Verify by thinking: number of permutations for 3 chosen (6 × 5 × 4 = 120) divided by 3! = 6 (orders of each team) → 120 ÷ 6 = 20 ✅
Quick Variations
1. Choosing r = 1 → nC1 = n (trivial).
2. Choosing r = n → nCn = 1 (only one way to choose all).
3. Use combinations for selecting committees, lottery picks, or unordered groups; use permutations if roles (order) are assigned within the selection.
Trick to Always Use
- Step 1: Ask: "Does order matter?" If NO → combinations (nCr).
- Step 2: Compute nCr either directly with factorials or as nPr ÷ r! for speed when r is small.
Summary
For selecting r items from n without caring about order:
- Use nCr = n! / (r! (n - r)!).
- Equivalently, compute permutations then divide by r! to remove ordering.
- Always confirm whether the problem requires order - that decides combination vs permutation.