COMBINATIONS nCr

COMBINATIONS nCr

What is it?

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.

In simple words

• 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.

Important terms

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.

What you should learn

• 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.

Examples

• 5C2=10
• nCn=1
• nC0=1

FORMULA

5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the group).

nCr = nC(n-r)

10C7 = 10C3. Use the smaller r to compute. Wording: 'committee', 'team', 'choose', 'hand of cards' -> combination.

Full school-level explanation

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.

Select, then the identity:

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

FORMULA

5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the group).

Remember

• divide out the orders

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.

Remember

• smaller r

Examples

• Choose 7 = leave 3
• Committee
• Not a ranked list

How to answer a combinations ncr question

How it works

• 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.

Remember

• order does not matter
• nCr = nPr / r!
• 56
• divide by r!
• divide out the orders
• smaller r
• order does not matter
• nCr = nPr / r!

How the ideas connect

Worked: 8C3. 8C3 = 8*7*63*2*1 = 3366 = 56

Deeper understanding

Go beyond memorising COMBINATIONS nCr. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: 8C3. 8C3 = 8*7*63*2*1 = 3366 = 56

Common mistakes

Exam focus

Core Idea

1. FORMULA

2. nCr = nC(n-r)

Worked Example

Exam focus

Quick revision

• order does not matter
• nCr = nPr / r!
• 56
• divide by r!
• COMBINATIONS nCr: A combination is a selection (order does not matter). nCr = n! / (r! (n-r)!) = nPr / r!. nC0 = 1, nC
• 1. FORMULA: 5C2 = 10. 10C3 = 120. Compute nPr then divide by r! (you overcounted by the r! orders inside the gro

nCr is smaller than nPr for 0<r<n because order is ignored.

For long answers on COMBINATIONS nCr: define clearly → explain the main parts → give one example → state one exam caution.