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Class 9 Combinatorics (Advanced): cheat sheet
CBSE Mathematics. Counting without listing: multiply for choices made one after another, add for choices that exclude each other, and use ⁿPᵣ or ⁿCᵣ.
Key ideas
- AND (one choice after another): multiply. OR (choices that cannot happen together): add.
- Use permutations when the order matters (seats, ranks, numbers, words) and combinations when it does not (teams, committees, lines, triangles).
- Fill a restricted place first, such as a first digit that cannot be 0 or a last digit that must be even.
- Objects that must stay together: treat them as one block, then multiply by the ways to arrange the block inside.
- "At least" problems: add the cases one by one, or take the unwanted cases away from the total.
- With m of n points on one line, those m give only 1 line and no triangles.
Remember
| Factorial | n! = n × (n − 1) × … × 2 × 1; 0! = 1 |
| Permutations | ⁿPᵣ = n! ÷ (n − r)!; with repetition: nʳ |
| Combinations | ⁿCᵣ = n! ÷ (r!(n − r)!); ⁿCᵣ = ⁿCₙ₋ᵣ |
| Lines and triangles, m of n points on a line | ⁿC₂ − ᵐC₂ + 1 lines; ⁿC₃ − ᵐC₃ triangles |
| Diagonals of an n-sided polygon | ⁿC₂ − n = n(n − 3)/2 |
| A telescoping sum | 1 × 1! + 2 × 2! + … + n × n! = (n + 1)! − 1 |
Watch out
- Using ⁿPᵣ for a committee, where the order of choosing does not matter.
- Letting 0 be the first digit of a number.
- Writing a! + b! = (a + b)!: 2! + 3! is 8, not 120.
Worked example
In how many ways can the letters of MOTHER be arranged so that (a) all the vowels come together (b) the vowels do not all come together?
Working: (a) 4 consonants + 1 vowel block: 5! × 2!; (b) 6! − (a) = 720 − 240
Answer: (a) 240; (b) 480
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