Worked solution
Divide by permutations of indistinguishable groups
Find the number of ways in which 6 people may be divided into groups of
(i) 1, 2 and 3,
(ii) 2, 2 and 2,
(iii) 3, 1, 1 and 1.
CJC · Discussion · Question 3
Answer
(i) 60; (ii) 15; (iii) 20.
(i) Different group sizes
Choose the singleton, then the pair, then the remaining triple: . Their sizes distinguish the groups, so no additional divisor is needed.
(ii) Three pairs
The sequential selection labels the first, second and third pairs artificially. Every unlabelled division occurs times, so the answer is .
(iii) One triple and three singletons
. Only the three singleton groups are indistinguishable; the triple has a different size.
Common mistakes
Do not divide by the factorial of all groups when only groups of equal size are indistinguishable.
Solution notes
All printed asks, working, counting factors and source annotations visually compared and retained. No marks are printed for this question.