Combinations | Moosa Academy

Combinations — Moosa Academy

A combination is a way of choosing a group from a larger set when the order does not matter. Picking Ahmed, Sara, and Mohammed for a team is the same choice however you list them.

Combinations Choosing a group

A combination chooses $r$ elements from $n$ with no regard to order, so $\{A, B, C\}$ and $\{C, B, A\}$ are the same combination — counted once.

Use combinations whenever you pick a group — a team, a committee, a hand of cards — where it does not matter who was chosen first.

Formula The combinations formula

C(n, r) = \frac{n!}{r!\,(n-r)!}

n

= the total number of elements,

r

= the number chosen.

Dividing by

r!

cancels the different orderings of the same group.

So a combination is a permutation with order removed:

C(n, r) = \dfrac{P(n, r)}{r!}

Example A 5-team league

Five teams, each pair playing once. Order does not matter, so count pairs:

C(5, 2) = \dfrac{5!}{2!\,3!} = \dfrac{120}{12}

⟹ 10 matches.

Example Three more

A committee of 4 from a class of 12

C(12, 4) = \dfrac{12!}{4!\,8!}

⟹ 495 ways.

Choosing 3 salads from 8 kinds

C(8, 3) = \dfrac{8!}{3!\,5!}

⟹ 56 choices.

A lineup of 5 players from 15

C(15, 5) = \dfrac{15!}{5!\,10!}

⟹ 3,003 different lineups.

Reference Combinations vs permutations

Property	Combinations C(n, r)	Permutations P(n, r)
Order	does not matter	matters
Formula	n! ÷ (r!(n − r)!)	n! ÷ (n − r)!
Result	smaller (or equal)	larger (or equal)
Example	choosing a team of 3	ranking 3 in positions
C(5, 2) vs P(5, 2)	10	20

Summary

A combination chooses a group with no regard to order: $C(n, r) = \dfrac{n!}{r!\,(n-r)!}$ .
Signal words like "group", "committee", or "team" mean you want combinations.
$C$ is smaller than $P$ : we divide by $r!$ to cancel the orderings of the same group.
A combination is a permutation with the order thrown away — $C(n, r) = P(n, r) \div r!$ .