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MathidiaMathematics Academy
Discrete mathematics
Lesson 46 of 75
Introduction

Counting principles

Objective. Use bijections, permutations and combinations.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Counting principles provide systematic tools for determining the number of ways events can occur, forming the basis of combinatorics. The multiplication principle states that if one task can be done in m ways and a second in n ways, both together can occur in mn ways; bijections then let complex counting problems be matched to simpler, already-solved ones. Permutations, nPr = n!/(n−r)!, count ordered arrangements of r items chosen from n, while combinations, nCr = n!/[r!(n−r)!], count unordered selections, dividing out the r! orderings that permutations distinguish. Choosing 2 letters from {A, B, C} in order gives 3P2 = 6 arrangements, while choosing them without regard to order gives 3C2 = 3 selections.

Worked example

A committee of 3 people is chosen from a group of 7 using combinations. Find the number of possible committees.

3 … 7
Illustrated card showing 3 … 7
  1. 1Recognise this as a combinations problem since committee order does not matter.
  2. 2Apply the formula nCr = n!/[r!(n−r)!] with n = 7 and r = 3.
  3. 3Compute the factorial expression: 7!/(3!4!) = (7×6×5)/(3×2×1).
  4. 4Simplify the numerator and denominator to find the total count.

Answer: 35

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Common mistake. Students use the permutation formula when order does not matter, overcounting by a factor of r!; always check whether the arrangement's order is relevant before choosing between nPr and nCr.

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