Counting Principles and Combinations

Explore multiplication principle, permutation count, combination count.

Lesson 1 of 53 parts Practice Flashcards

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Multiplication Principle

Objective: Explain multiplication principle and use it to solve or interpret a problem.

If one stage has m options and an independent next stage has n options, the total ordered outcomes are mn. Check that choices really form separate stages. When an earlier selection changes later options.

Worked examples

Try itCounting Principles and Combinations — practice 1.1: How many ordered ways can 2 roles be assigned among 7 people?Show answer

Step 1: Choose the first role from 7 people. Step 2: Then choose the second from 6 remaining. Step 3: Multiply to get 42.

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Multiplication Principle

Objective: Explain multiplication principle and use it to solve or interpret a problem.

Worked examples

Try itCounting Principles and Combinations — practice 1.1: How many ordered ways can 2 roles be assigned among 7 people?Show answer

Step 1: Choose the first role from 7 people. Step 2: Then choose the second from 6 remaining. Step 3: Multiply to get 42.

Permutation Count

Objective: Explain permutation count and use it to solve or interpret a problem.

An arrangement assigns order or roles, so selecting A then B differs from B then A. Count without replacement using falling products. They assign people to different roles or positions.

Worked examples

Try itCounting Principles and Combinations — practice 2.1: How many ordered ways can 2 roles be assigned among 10 people?Show answer

Step 1: Choose the first role from 10 people. Step 2: Then choose the second from 9 remaining. Step 3: Multiply to get 90.

Combination Count

Objective: Explain combination count and use it to solve or interpret a problem.

A combination selects a group without regard to order; divide ordered counts by the number of arrangements of each selected group. When only membership matters, not order.

Worked examples

Try itCounting Principles and Combinations — practice 3.1: How many unordered pairs can be chosen from 13 people?Show answer

Step 1: Count ordered pairs: 156. Step 2: Divide by 2! because pair order does not matter. Step 3: The answer is 78.

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