Permutations and Combinations
What the exam tests
This domain tests whether an arrangement (order matters) or a selection (order does not matter) is required. Typical items involve committees, seating, word arrangements with repeated letters, and “at least” conditions.
Core rules
- ({}^nP_r=\frac{n!}{(n-r)!}), the number of ordered arrangements of (r) objects from (n).
- ({}^nC_r=\frac{n!}{r!(n-r)!}), the number of selections. Also ({}^nP_r=r!,{}^nC_r).
- ({}^nC_0={}^nC_n=1), ({}^nC_r={}^nC_{n-r}), and ({}^{n}C_r+{}^{n}C_{r-1}={}^{n+1}C_r).
- Arrangements of (n) objects with repeats (p,q,\dots) are (\frac{n!}{p!q!\cdots}).
- Circular arrangements of (n) distinct objects are ((n-1)!), or (\frac{(n-1)!}{2}) when clockwise and anticlockwise are the same.
Worked example
From 5 men and 4 women, form a committee of 3 containing at least one woman. Total committees: ({}^{9}C_3=84). Committees with no woman are all-male: ({}^{5}C_3=10). By complement, the required number is (84-10=74). The order of committee members is irrelevant, so combinations-not permutations-are used.
Traps
“Arrange”, “seat” and “rank” usually signal permutations; “select”, “committee” and “group” signal combinations. For “at least one”, count the complement when it is simpler. Divide by factorials for repeated letters. Do not use ((n-1)!) for a straight-line arrangement. Check whether a circular table has distinguishable clockwise/anticlockwise orders.