MathematicsChapter 6: Permutation and Combination
COMBINATIONS nCr
A combination is a selection (order does not matter).
Learning objectives
- State the main ideas and notation used in combinations ncr.
- Apply the rules of combinations ncr in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A combination is a selection (order does not matter). nCr = n! / (r! (n-r)!) = nPr / r!. nC0 = 1, nCn = 1, nC1 = n. nCr = nC(n-r) (choosing r to take is choosing n-r to leave). Meaning: a committee of 3 from 10 is 10C3, not 10P3, because the three seats are identical.
Mathematical form
nCr = n
nC0 = 1
nCn = 1
nC1 = n. nCr
Where / symbols
- COMBINATIONS nCr
- A combination is a selection (order does not matter). nCr = n! / (r! (n-r)!) = nPr / r!. nC0 = 1, nCn = 1, nC1 = n. nCr = nC(n-r) (choosing r to take is choosing n-r to leave). Meaning: a committee of 3 from 10 is 10C3, not 10P3, because the three seats are identical.
- 1. FORMULA
- 5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the group).
- 2. nCr = nC(n-r)
- 10C7 = 10C3. Use the smaller r to compute. Wording: 'committee', 'team', 'choose', 'hand of cards' -> combination.
Relationship
- nCr = nPr / r!
Overview
Select, then the identity:
Formula
5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the group).
Examples 5C2=10 · nCn=1 · nC0=1
nCr = nC(n-r)
10C7 = 10C3. Use the smaller r to compute. Wording: 'committee', 'team', 'choose', 'hand of cards' -> combination.
Examples Choose 7 = leave 3 · Committee · Not a ranked list
WHAT — COMBINATIONS nCr
A combination is a selection (order does not matter). nCr = n! / (r! (n-r)!) = nPr / r!. nC0 = 1, nCn = 1, nC1 = n. nCr = nC(n-r) (choosing r to take is choosing n-r to leave). Meaning: a committee of 3 from 10 is 10C3, not 10P3, because the three seats are identical.
RULE — 1. FORMULA
5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the group).
METHOD — 2. nCr = nC(n-r)
10C7 = 10C3. Use the smaller r to compute. Wording: 'committee', 'team', 'choose', 'hand of cards' -> combination.
WORKED EXAMPLE
Worked: 8C3. 8C3 = 8*7*63*2*1 = 3366 = 56
How to answer a combinations ncr question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- 8C3.
- 8C3 = 8*7*63*2*1
- = 3366
- = 56
In simple words
nCr is smaller than nPr for 0<r<n because order is ignored.
Key points
- order does not matter
- nCr = nPr / r!
- divide out the orders
- smaller r
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.COMBINATIONS nCr
Sticky Notes · Quick Revision
Core Idea
- A combination is a selection (order does not matter).
- nCr = n!
- / (r!
1. FORMULA
- 5C2 = 10.
- 10C3 = 120.
- Compute nPr then divide by r!
- (you overcounted by the r!
2. nCr = nC(n-r)
- 10C7 = 10C3.
- Use the smaller r to compute.
- Wording: 'committee', 'team', 'choose', 'hand of cards' -> combination.
Worked Example
- Worked: 8C3.
- 8C3 = 8*7*63*2*1
- = 3366
- = 56
Common Misconception
- nCr is smaller than nPr for 0<r<n because order is ignored.
FBISE Exam Check
- State the rule for combinations ncr before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A combination is a selection (order does not matter).
- nCr = n!
- / (r!
- nCr is smaller than nPr for 0<r<n because order is ignored.
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