Arrangement vs selection
Permutation = arrangement (order matters). Combination = selection (order does not matter). Ask yourself: "If I swap two chosen items, do I get a new outcome?" Yes means permutation. Target: 45 seconds.
Key formulas
| Concept | Formula |
|---|---|
| n! | n x (n - 1) x ... x 1; 0! = 1 |
| nPr | n!/(n - r)! |
| nCr | n!/(r! (n - r)!); nCr = nC(n - r) |
| Letters with repeats (p alike, q alike) | n!/(p! q!) |
| Circular arrangement | (n - 1)! |
| Necklace / garland (flip allowed) | (n - 1)!/2 |
| Diagonals of an n-gon | nC2 - n = n(n - 3)/2 |
| Triangles from n points (no 3 collinear) | nC3 |
Fundamental principles
- AND means multiply (choose a shirt and a trouser).
- OR means add (travel by bus or by train).
- At least questions: total minus the unwanted case, or add each valid case.
Word arrangements
- Vowels together: tie the vowels into one block, arrange the blocks, then multiply by arrangements inside the block.
- Vowels never together: total minus together.
- Vowels at odd places: choose the odd places for vowels first.
- No two girls together: arrange the boys, then place girls in the gaps (boys + 1 gaps).
Digits
For numbers, the first digit cannot be 0. For divisibility by 5, fix the last digit as 0 or 5 and count the two cases separately.
Worked example
Arrangements of the word LETTER: 6 letters with T twice and E twice = 720/(2 x 2) = 180.
Worked example 2
A committee of 3 from 5 men and 4 women with exactly one woman: 4C1 x 5C2 = 4 x 10 = 40.
Worked example 3
6 people around a table with two particular people together: (5 - 1)! x 2 = 48.
Common traps
- Using permutation for a committee (selection).
- Forgetting to divide by factorials of repeated letters.
- Allowing 0 as the first digit of a number.
- "At least one" counted case by case but missing a case.
Worked example 4 (same colour)
Two balls are selected from 4 red and 3 blue balls so that both are the same colour: 4C2 + 3C2 = 6 + 3 = 9 ways (OR means add).
Worked example 5 (fixed first letter)
Arrangements of the letters of CLERK that begin with C: the remaining 4 letters can be arranged in 4! = 24 ways.