Permutations and Combinations
Permutations and Combinations tests counting arrangements and selections, restrictions, circular permutations and combinations with conditions.
Core rules
- If order matters, count permutations: nPr=n!/(n-r)!. If order does not matter, count combinations: nCr=n!/(r!(n-r)!).
- nCr=nC(n-r); nC0=nCn=1.
- Circular arrangements of n distinct objects number (n-1)!. If clockwise and anticlockwise are the same, divide by 2.
- For at least one condition, use total minus none; for adjacent objects, treat the block as one unit and arrange inside the block.
Worked example
From 5 men and 4 women, choose a committee of 3 with at least one woman. Total committees are 9C3=84. Committees with no woman are 5C3=10. Therefore the required number is 84-10=74. The complement avoids separate cases of one, two and three women.
Traps
- Do not use nPr when the selected group is unordered; it overcounts by r!.
- For repeated letters, divide by the factorial of each repeated count.
- In circular permutations, fixing one object removes the rotational symmetry; do not subtract 1 after arranging.
- At least one is usually easier by complement, but exactly one requires direct cases.
- Check whether people or objects are distinct before using n!.