Permutations & Combinations

Formula Sheet 9

Definition:

The product of all positive integers from 1 to n.

$$ n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1 $$

Crucial Rule:

$$ 0! = 1 $$

Permutation (Arrangement)	Combination (Selection)
Order Matters $$ ^nP_r = \frac{n!}{(n-r)!} $$ Used for: Arranging people in a row, ranking, digits in a number.	Order Doesn’t Matter $$ ^nC_r = \frac{n!}{r!(n-r)!} $$ Used for: Selecting a team, picking cards, choosing colors.

Permutation (Arrangement)

Combination (Selection)

Order Matters

$$ ^nP_r = \frac{n!}{(n-r)!} $$

Used for: Arranging people in a row, ranking, digits in a number.

Order Doesn’t Matter

$$ ^nC_r = \frac{n!}{r!(n-r)!} $$

Used for: Selecting a team, picking cards, choosing colors.

Symmetry Rule:

$$ ^nC_r = ^nC_{n-r} $$

Boundary Values:

$$ ^nC_0 = 1 $$ and $$ ^nC_n = 1 $$

Relation:

$$ ^nP_r = r! \times ^nC_r $$