Factorials and combinations
Factorials multiply consecutive positive integers. Combinations count unordered selections, so repeated orders are divided out
\[\binom{n}{r}=\frac{n!}{r!\left(n-r\right)!}\]
- Use the factorial definition A factorial is the product of all positive integers down to 1, and 0! equals 1 \(\displaystyle 0!=1,\quad4!=4\times3\times2\times1\)
- Cancel consecutive factorial factors Expand only the factors needed to cancel a factorial in the denominator \(\displaystyle \frac{8!}{5!}=8\times7\times6\)
- Use combinations for unordered groups A combination counts a selected group once when its order does not matter \(\displaystyle \binom{8}{3}=\frac{8!}{3!5!}\)
- Keep order only when positions matter An ordered selection uses successive choices without dividing out their rearrangements \(\displaystyle P\left(8,3\right)=8\times7\times6\)
Worked example
Evaluate the combination below
\[\binom{9}{2}\]
- \(\displaystyle 9-2=7\)The number of unselected objects is 7
- \(\displaystyle \binom{9}{2}=\frac{9!}{2!7!}=\frac{9\times8}{2\times1}\)Use the combination formula and cancel the 7! factor
- \(\displaystyle \frac{9\times8}{2\times1}=36\)Evaluate the remaining product to obtain 36
Common mistakes
Treating a combination as ordered
✕\(\displaystyle \binom{8}{3}=8\times7\times6=336\)✓\(\displaystyle \binom{8}{3}=\frac{8\times7\times6}{3\times2\times1}=56\)
A combination counts one selected group, so repeated orders must be divided outDistributing a factorial over a sum
✕\(\displaystyle \left(5+2\right)!=5!+2!=122\)✓\(\displaystyle \left(5+2\right)!=7!=5040\)
The factorial applies to the total sum before the factorial is evaluatedUsing an incomplete combination denominator
✕\(\displaystyle \binom{9}{3}=\frac{9!}{3!\times6}=10080\)✓\(\displaystyle \binom{9}{3}=\frac{9!}{3!6!}=84\)
The denominator contains the factorial of both the selected count and the unselected countDifficulty levels
introThis tier evaluates small factorials and direct combinations
standardThis tier evaluates factorial ratios and unordered selections
challengeThis tier evaluates composed expressions with restrictions on the selection values
examThis tier distinguishes ordered from unordered selections before evaluating exactly
Questions about this topic
Why is 0! equal to 1?+
It is the multiplicative identity required by the factorial recurrence and by the combination formula
When is a combination used instead of an arrangement?+
Use a combination when changing the order of the selected objects does not create a new outcome
How can a factorial ratio be evaluated efficiently?+
Expand only the consecutive factors that do not cancel, then simplify before multiplying
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 11: Binomial distribution
In this chapter
Bernoulli trials and the Bernoulli distribution11A Bernoulli trials and the Bernoulli distribution: Finite events, probability assignments and complementsSolve a quadratic exactly with the quadratic formula11A Bernoulli trials and the Bernoulli distribution: Round to decimal places and significant figures11B Binomial distribution and parameters: Binomial conditions, the distribution table and graph shapeMean, variance, standard deviation and parameter recovery of a binomial variableExact probability of repeated independent successesEvaluate factorial and combination expressionsSolve a linear equation11B Binomial distribution and parameters: Round to decimal places and significant figuresCumulative, at-least-one and conditional probabilities of a binomial variableSmallest trial count for an at-least-one threshold11C Binomial distribution events: Finite events, probability assignments and complementsConditional probability from exact representationsSolve and interpret a linear inequation11D Binomial shape and specified order: Binomial conditions, the distribution table and graph shapeProbabilities of sequences of independent eventsDescribe the shape of a distribution