Permutations

Permutations are used when selecting objects with respect to the order.

If we have a set of $n$ objects and we want to choose $r$ objects from the set in order, we write $P(n,r)$ or $_nP_r$.

To calculate $P(n,r)$, we find the factorial $n!$ i.e. the number of ways to line up all $n$ objects. Then we divide it by $(n-r)!$ to cancel out the $(n−r)$ items that we do not wish to line up.

Given $n$ distinct objects, the number of ways to select $r$ objects from the set in order is:

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

Example: n=6, r=3

$$\frac{n!}{(n-r)!} = \frac{6!}{3!} = \frac{6·5·4· \not 3!}{\not 3!} = 6·5·4 = 120$$

Results of permutations are sequences (ordered lists).

Results of combinations are sets.

Examples

A horse race with 10 horses

Select 3 winners (in any order):

C(10,3) "10 choose 3":

n!/r!(n-r)!
10!/3!(10-3)!
= 10*9*8*7!/3!*7!
= 10*9*8/3!
= 720/6
= 120

i.e. there are 120 different 3-horse subsets that can be formed out of the 10-horse set. If you've placed 1 bet, then you've chosen 1 out of 120 distinct possibilities.

Select 3 winners in exact order is P(10,3):

n!/(n-r)!
10!/(10 - 7)!
= 10!/7!
= 10*9*8*7!/7!
= 10*9*8
= 720

How many possible 5-card hands

The number of distinct 5-card subsets out of 52-card set.

C(52,5)
= 52!/5!*47!
= 52*51*50*49*48*47!     / 5!*47!
= 52*51*50*49*48         / 5!
= 52*51*50*49*(12*4)     / 120
= 52*51*(5*10)*49*(12*4) / 10*12
= 52*51*5*49*4
= 52*51*49*20
= 2,598,960

How many distinct full-house hands

Full house is 3 of a kind and a pair

• 13 ranks: 2-10,J,Q,K,A

• Choose denominaton: C(13,1) = 13

• For each denomination choose 3 of a kind out of 4 of a kind

• 4-choose-3, C(4,3) = 4

• The number of ways tochoose 3 of a kind: 13*4=52

• 12 ranks remaining

• C(12,1) = 12

• Choose a pair

• C(4,2) = 6

• remaining pairs: 12*6=72

• Total number of distinct full-house hands: 52*72 = 3744

What is the probability of being delt a fullhouse

nuts (no need to discard and draw) considering the classic 5-card draw poker?

• number of full-house hands: 52*72 = 3744

• number of hands: 52-choose-5 = 2,598,960

• probability: 3,744/2,598,960 = 0.00144 i.e. 0.144%

• which also means you get a full house straight up every 695 rounds (average)

The law of large numbers If an event has even a slim a chance of happening (aslong asit is not none), it will certainly happen given sufficently large time (years, units, etc.).