Combinations

ACT Statistics and Probability

In this lesson:

• Understand why combinations count less than permutations
• Apply the formula
• Decide: permutation or combination?

Your Learning Goals for This Lesson

1. Solve counting problems (committees, teams, hands): decide whether order matters, count with , multiply stage counts
2. Decide permutation or combination: does order matter?
3. Compute
4. Explain why

Listing Permutations and Grouping Them

Recall: and . Choose 2 from :

AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC

Group pairs with the same letters: {AB,BA} {AC,CA} {AD,DA} {BC,CB} {BD,DB} {CD,DC}

6 groups, so .

Why Combinations Are Smaller Than Permutations

Each combination of items has orderings, all counted in .

For the example: .

For the same and , .

Second Case: Choose Three From Five

From : ordered selections.

Each group of 3 has orderings, for example ABC, ACB, BAC, BCA, CAB, CBA.

Quick Check: Overcounting Groups of Three Items

If you choose 3 items from a set, each combination appears how many times in the list of permutations?

Think before advancing...

Combination Formula with Annotated Parts

is the pool and the number chosen, so .

Cancel Before You Multiply Factors

1. Use the smaller of and : .
2. Write the top factors of over .
3. Cancel first. A committee of 4 from 10 volunteers:

Example: Pizza Toppings, Choose Three of Eight

8 toppings, choose 3. Pepperoni-olive-mushroom is the same pizza as olive-mushroom-pepperoni.

• ,

Permutation vs Combination: The Decision

Order MATTERS → Permutation : ranked positions, assigned roles

Order does NOT matter → Combination : committees without roles, card hands, menu choices

Test: "Would rearranging the selection give a different outcome?"

President and vice-president from 8: , not .

Same Context, Two Formulas: Compare Results

12 applicants, 5 selected for a team:

Same and , different formulas, very different answers.

Mixed Problem: Lottery Ticket in Three Steps

Choose 5 numbers from 1 to 40. How many tickets?

1. Order matter? No: the same 5 numbers are the same ticket.
2. , .

Mixed Practice: Three Counting Problems

Decide, then solve.

1. 9 students; 3 form a study group. Groups?
2. 7 runners; medals for 1st, 2nd, 3rd. Outcomes?
3. 6 flavors; choose 2 different flavors. Pairs?

Mixed Problem: Multi-Stage Counting With Combinations

Recall: 3 shirts, 2 pants: outfits.

Choose a subcommittee of 3 from 8 people, then a chair from those 3. How many outcomes?

(a) (b) (c) (d)

Stages: choose the 3, then the chair: , choice (b).

Key Takeaways: Combination Rules and Warnings

• , with two factorials in the denominator
• Order does not matter: combination. Order matters: permutation
• always
• Staged selection: count each stage, multiply

Warnings: keep both and ; cancel before multiplying; assigned roles mean permutation

Coming Up Next in Statistics and Probability

Up next: Sampling methods

• Random, stratified, cluster, and convenience samples
• How each method picks a sample
• Telling the methods apart