- LPU MCA Semester 1
- PEA515
- Unit 5
PEA515 · Unit 5: Permutation, combination and probability
Selection and arrangement problems: numbers, words and geometric applications
30 practice questions · in the ETE syllabus
Key points
- Permutation if order matters; combination if it does not
- Repeated letters: divide by factorials of repeats
- Together: glue and swap; apart: total minus together
- At least one = total minus none
A permutation is an arrangement in which order matters: P(n, r) = n! / (n - r)!. A combination is a selection in which order does not matter: C(n, r) = n! / (r! (n - r)!).
| Problem | Count |
|---|---|
| Posts filled one by one (president, secretary...) | P(n, r) |
| Committee of r from n | C(n, r) |
| At least one woman | All committees minus those with no woman |
| Exactly j women and the rest men | C(w, j) x C(m, k - j) |
| Letters of a word with repeats | n! / (p! q! ...) for repeat counts p, q |
| Two particular people together in a row | 2 x (n - 1)! (glue them, then swap) |
| Two particular people not together | n! - 2 x (n - 1)! |
Example: the letters of SUCCESS (S three times, C twice) can be arranged in 7! / (3! x 2!) = 420 ways. Selecting 4 from 3 men and 5 women with at least one woman gives C(8, 4) - C(3, 4) = 70 - 0 = 70.
Exam trap: when two people must be together, forgetting that they can swap places halves the answer.
Try two questions
A committee of 5 is to be formed from 4 men and 5 women. In how many ways can it be formed so that it has at least one woman?
- 350
- 0
- 5
- 126
Show the answer
D. All committees C(9, 5) minus those with no woman C(4, 5) = 126.
Why not A: Counts selections with several women more than once.
Why not B: Counts committees with no woman.
Why not C: Counts only committees with exactly one woman.
A committee of 4 is to be formed from 6 men and 2 women. In how many ways can it be formed so that it has at least one woman?
- 70
- 15
- 40
- 55
Show the answer
D. All committees C(8, 4) minus those with no woman C(6, 4) = 55.
Why not A: Ignores the condition.
Why not B: Counts committees with no woman.
Why not C: Counts only committees with exactly one woman.