Combination casework
A restriction on an unordered group changes which selections are legal. Translate the words into category counts before using nCr.
Choose 3 with at least 2 violinists
case 1
2 violinists and 1 guitarist
⁵C₂ × ⁴C₁ = 40
case 2
3 violinists
⁵C₃ = 10
40 + 10 = 50 selections
Multiply within one case; add separate cases
A case is one possible make-up of the group. Separate cases must not overlap: one selection cannot belong to two cases that you add.
Worked example
At least two violinists
Choose 3 musicians from 5 violinists and 4 guitarists.
- Two violinists and one guitarist: .
- Three violinists and no guitarist: .
- Add the two non-overlapping case totals.
: there is one way to select nobody from a category. “At least” or “at most” usually needs a list of cases.
Remove compulsory or forbidden members first
Worked example
Zara must be on the committee
A four-person committee is selected from 9 people.
- Place Zara in the committee.
- Choose the other 3 members from the remaining 8 people.
Key idea
Check whether the groups have names
- For named groups or groups of different sizes, choose the members of each group in turn. The final group is forced.
- Equal-sized groups with no names can be swapped without creating a new division. Divide by the factorial of the number of interchangeable groups.
Worked example
Three different group sizes
Divide 11 students into groups of 6, 3 and 2.
The sizes identify the groups, so swapping whole groups is not a new issue.
Six students are divided into two unnamed groups of three. How many different divisions are possible?
Show worked answer
Choosing the first group also chooses the second. Each division is counted twice because the two group names do not matter.