Search topics

Search

Evaluate factorial and combination expressions

Start a set of 8
Step 1 of 8

Factorials count orders of distinct objects

A factorial multiplies each positive integer down to one. The empty set has exactly one ordering, the empty list. Factorials here do not take negative integer arguments. Cancel shared factors before multiplying large products

\[n!=n(n-1)\cdots1\quad(n\ge1),\qquad 0!=1\]
Year 11, VCE Units 1 & 2, Chapter 4, 3 difficulty levels, No calculatorBuild a worksheet

Factorials and combinations

Specify whether positions or only membership matter

  1. Ordered, without repetition For a valid selection size \(\displaystyle \frac{n!}{(n-k)!},\qquad 0\le k\le n\)
  2. Unordered, without repetition Count each selected group once \(\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}\)
  3. Boundary The empty and complete selections each have one outcome \(\displaystyle \binom{n}{0}=\binom{n}{n}=1,\qquad 0!=1\)

Worked example

Compare assigning two different roles to nine people with selecting an unlabelled pair

\[{}_9P_2=9\times8\]
  1. \(\displaystyle {}_9P_2=72\)Role order matters
  2. \(\displaystyle \binom92=\frac{72}{2!}=36\)Each pair has two role assignments

Common mistakes

Dividing when the roles matter
✕\(\displaystyle \binom{n}{2}\)✓\(\displaystyle n(n-1)\)
An ordered pair is different when its positions are swapped

Difficulty levels

introSmall factorial and selection counts
standardCancel factorial ratios and evaluate combinations
challengeUse a required-object condition and composed ratios

Questions about this topic

Does C(n,k) allow repeated objects?+

No. It selects k different members of an n-element set. Unlimited repeated types need a different count

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 4: Discrete mathematics

In this chapter

Find term n of an arithmetic sequence using the ruleFind term n of a geometric sequence using the ruleFinite sums and geometric limitsIterate a first-order recurrenceExplicit recurrence solutions, equilibria and population modelsModel and interpret a periodic cash-flow recurrenceEvaluate factorial and combination expressionsCount arrangements with product, sum and restriction rulesCount restricted selections and calculate exact probabilitiesPigeonhole bounds, overlap counts and repeated selectionsRead, transpose and classify a matrixCombine scalar multiples of matricesMultiply matricesDetermine whether an inverse exists and calculate itSolve a matrix equationSolve simultaneous equations using the inverse matrixUse matrix multiplication to answer an application question