Permutations and Combinations

$n! = n(n-1)(n-2)...(3)(2)(1)$

Example 2:

a) $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

b) $9! = 362,880$

c) $\frac{4!}{2!3!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} = 2$

Example 3: 8 locations can be ranked in $8! = 40,320$ ways. The top 4 can be ranked in $8 \times 7 \times 6 \times 5 = 1680$ ways.

A permutation is an arrangement of objects in a specific order.

Permutation of n objects taking r at a time ($r \le n$):

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

Example 4:

a) $P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$

b) $P(10, 8) = \frac{10!}{(10-8)!} = \frac{10!}{2!} = 1,814,400$

Example 5: 8 students on 6 chairs: $P(8, 6) = \frac{8!}{2!} = 20,160$ ways.

Permutation of n objects taking n at a time:

$P(n, n) = n!$

Example 6: 5 books arranged 5 at a time: $P(5, 5) = 5! = 120$ ways.

Permutations with alike objects:

If there are n objects with r alike objects of one kind and q alike objects of another kind, the number of permutations is:

$\frac{n!}{r!q!}$

Example 7: "MOROGORO" has 8 letters, 4 "O"s, and 2 "R"s: $\frac{8!}{4!2!} = 840$ patterns.

A combination is a selection of objects without regard to order.

Combination of n objects taking r at a time ($r \le n$):

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

Relations of combinations:

1. $C(n, r) = C(n, n-r)$
2. $C(n, n) = 1$
3. $C(n, 0) = 1$
4. $r! \times C(n, r) = P(n, r)$

Example 8: 4 people from a group of 10: $C(10, 4) = \frac{10!}{6!4!} = 210$ committees.

Example 9: 4 letters from 6: $C(6, 4) = \frac{6!}{2!4!} = 15$ selections.