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

By the end of this lesson, you will be able to:

1. Explain why for the same and
2. Apply to count unordered selections
3. Choose permutations vs combinations for each ACT problem
4. Solve problems involving committees, teams, and card hands

Listing Permutations and Grouping Them

Choose 2 from . All ordered pairs (12 total):

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

Group by same letters: {AB,BA} {AC,CA} {AD,DA} {BC,CB} {BD,DB} {CD,DC} → 6 groups

Each group = one combination. .

Why Combinations Are Smaller Than Permutations

Each combination of items generates permutations.

For the example:

Dividing by removes the overcounting from ordering.

Worked Example: Confirming

ordered arrangements.

Each pair appears twice (AB and BA):

Verify by listing: AB, AC, AD, BC, BD, CD — exactly 6 unordered pairs. ✓

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

= total items; = items chosen (order does not matter)

Example 1: Committee of 4 from 10

A committee of 4 from 10 volunteers. Order does not matter.

• ,

Example 2: Pizza Toppings —

8 toppings; choose 3. Pepperoni-olive-mushroom = olive-mushroom-pepperoni.

• ,

Quick Check: Compute the Combination

Show each step before advancing…

Permutation vs Combination: The Decision

Order MATTERS → Permutation :

• Ranked positions: 1st, 2nd, 3rd
• Assigned roles: president, VP, secretary

Order does NOT matter → Combination :

• Committees and teams without roles
• Lottery numbers, card hands, menu choices

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

Same Context, Two Formulas: Compare Results

12 applicants, 5 selected for a team:

Same and — different formulas — very different answers.

ACT Language Cues for Combination Problems

Phrases that signal a combination:

• "committee" / "team" → no distinct roles
• "choose" / "select" / "pick" → unordered group
• "lottery numbers" → drawn set, not sequence
• "hand of cards" → group, not ranked order

Strategy: Underline the group-selection phrase. If no roles assigned, use .

ACT Problem 1: Lottery Ticket

Choose 5 numbers from 1–40 for a lottery ticket. How many tickets exist?

• Order does not matter (same 5 numbers = same ticket)
• ,

ACT Problem 2: Hand of Cards

5-card hand from a 52-card deck. How many distinct hands?

• Order does not matter (same 5 cards = same hand)
• ,

ACT Practice: Three Counting Problems

Decide: permutation or combination? Then solve.

1. 9 students; 3 form a study group. How many groups?
2. 7 runners; medals for 1st, 2nd, 3rd place. How many outcomes?
3. A shop has 6 flavors; choose 2 scoops (different flavors). How many pairs?

Check Your Work: Practice Solutions

1. Group, no roles →

2. Ranked medals →

3. Unordered pair of flavors →

Problem 1 and 3 use combinations; Problem 2 uses permutations.

Key Takeaways: Combination Rules and Warnings

• — two factorial terms in denominator
• Order does not matter → combination; order matters → permutation
• always

Denominator needs both and — never drop one

Write top numerator factors, then divide by

What You Will Learn Next

Future topics build on both counting techniques:

• Multi-step counting: combine multiplication with and
• Probability with combinations:
• Pascal's triangle and its connection to

Permutations and combinations are the foundation for probability calculations.