8 min readConcept sheetShort calculations; a 10-mark question with several parts

Permutations and combinations

Use a permutation when order matters, as in arrangements and rankings. Use a combination when order does not matter, as in choosing a team.

n factorialn! = n × (n − 1) × … × 1; 0! = 1

PermutationsnPr = n! ÷ (n − r)!

CombinationsnCr = n! ÷ [r!(n − r)!]

LinknPr = nCr × r!

Does order matter?

Yes: permutationArranging, ranking, codes, seating, president-secretary-treasurer. AB and BA are different.

No: combinationChoosing, selecting, committees, teams, a hand of cards. AB and BA are the same selection.

Quick test: swap two chosen items. If the result is a different outcome, order matters.

Counting rules

Multiplication principlem ways then n ways gives m × n ways

n different objects in a rown!

Arrange r of n different objectsnPr = n! ÷ (n − r)!

With repeats (p alike, q alike)n! ÷ (p! q!)

n objects round a table(n − 1)!

Choose r of nnCr = n! ÷ [r!(n − r)!]

SymmetrynCr = nC(n − r); nC0 = nCn = 1

Arrangements (order matters)

Problem

(a) In how many ways can 5 people sit in a row? (b) In how many ways can a president, secretary and treasurer be chosen from 10 members? (c) How many arrangements are there of the letters of COLOMBO?

Working

(a) 5! = 5 × 4 × 3 × 2 × 1 = 120.
(b) Order matters (the posts differ): 10P3 = 10 × 9 × 8 = 720.
(c) COLOMBO has 7 letters with O repeated 3 times.
Arrangements = 7! ÷ 3! = 5,040 ÷ 6 = 840.

Answer

(a) 120 (b) 720 (c) 840

Selections (order does not matter)

Problem

(a) How many committees of 3 can be chosen from 10 people? (b) A committee of 2 men and 3 women is chosen from 6 men and 5 women. How many ways?

Working

(a) 10C3 = 10! ÷ (3! × 7!) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.
(b) Men: 6C2 = (6 × 5) ÷ 2 = 15. Women: 5C3 = (5 × 4 × 3) ÷ 6 = 10.
Both choices are made, so multiply: 15 × 10 = 150.

Answer

(a) 120 (b) 150

PIN codes (multiplication principle)

Problem

How many 4-digit PINs can be made from the digits 0-9 (a) if digits may repeat, (b) if no digit repeats?

Working

(a) Each of the 4 places has 10 choices: 10 × 10 × 10 × 10 = 10⁴ = 10,000.
(b) 10 choices, then 9, then 8, then 7: 10 × 9 × 8 × 7 = 10P4 = 5,040.

Answer

(a) 10,000 (b) 5,040

In Sri Lankan business

A Sri Lankan bank counting how many 4-digit PINs exist, or a company board choosing 3 directors from 8 nominees (8C3 = 56 ways), is using permutations and combinations.

Exam angle

Short answer: “Evaluate 8C3 and 8P3.” 8C3 = 56; 8P3 = 8 × 7 × 6 = 336.

Structured question: “How many arrangements of the letters of COLOMBO are there?” 7 letters, O three times: 7! ÷ 3! = 840.

Essay: “A committee of 5 is chosen from 6 men and 5 women. In how many ways if it has (a) exactly 2 women, (b) at least 1 woman?” (a) 5C2 × 6C3 = 200. (b) 11C5 − 6C5 = 462 − 6 = 456.

Remember it

Permutation = Position matters. Combination = Choose a group. And nPr = r! × nCr, so nPr is bigger whenever r ≥ 2.