Factorials and combinations
Specify whether positions or only membership matter
- Ordered, without repetition For a valid selection size \(\displaystyle \frac{n!}{(n-k)!},\qquad 0\le k\le n\)
- Unordered, without repetition Count each selected group once \(\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}\)
- 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
\[{}_9P_2=9\times8\]
- \(\displaystyle {}_9P_2=72\)Role order matters
- \(\displaystyle \binom92=\frac{72}{2!}=36\)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 swappedDifficulty 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