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# Combinations and Permutations

## Key Questions

See below:

#### Explanation:

A combination is a grouping of distinct objects without regard to the order in which the grouping is made.

As an example, a poker hand is a combination - we don't care in what order we're dealt the cards, only that we're holding a Royal Flush (or a pair of 3s).

The formula for finding a combination is:

C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!) with $n = \text{population", k="picks}$

As a for instance, the number of possible 5-card poker hands is:

C_(52,5)=(52!)/((5)!(52-5)!)=(52!)/((5!)(47!))

Let's evaluate it!

(52xx51xxcancelcolor(orange)(50)^10xx49xxcancelcolor(red)48^2xxcancelcolor(brown)(47!))/(cancelcolor(orange)5xxcancelcolor(red)(4xx3xx2)xxcancelcolor(brown)(47!))=52xx51xx10xx49xx2=2,598,960

• Permutation involves arrangements whereas combination implies number of selections or combinations.
Number of permutations of three letters from given five letters a, b, c, d, e is 60 as follows
abc,acb,bac,bca,cab,cba
abe,...
acd,..
ace,...
bcd,..
bce,...
bde,...
cde,...
10 x 6 = 60
now number of combinations of five letters a,b,c,d,e taken three at a time is just ten, namely

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