# Combinations and Permutations

Fundamental Counting Principle Probability

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## Key Questions

• The counting principle has 2 parts: the multiplication principle and the addition principle.

The multiplication principle says that if you know the number of ways one event could occur and the number of ways another event could occur, you can determine the number of ways both events could occur together by multiplying.

The addition principle says that if you have a situation where one event could occur or another event could occur, you can determine the number of ways that one or the other could occur by adding.

• A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can arrange 2 letters from that set. Each possible arrangement would be an example of a permutation.

The complete list of possible permutations would be:

$A B , A C , B A , B C , C A , \mathmr{and} C B$.

When they refer to permutations, statisticians use a specific terminology. They describe permutations as n distinct objects taken r at a time.

Translation: n refers to the number of objects from which the permutation is formed; and r refers to the number of objects used to form the permutation.

Consider the example from the previous paragraph. The permutation was formed from 3 letters (A, B, and C), so n = 3; and the permutation consisted of 2 letters, so r = 2.

Computing the number of permutations

The number of permutations of n objects taken r at a time is:

nPr = n(n - 1)(n - 2) ... (n - r + 1) = (n!) / ((n - r)!)

• A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC.

Computing the number of combinations. The number of Combinations of n objects taken r at a time is
nCr = n(n - 1)(n - 2) ... (n - r + 1)/r! = n! / r!(n - r)! = nPr / r!

• 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

• Permutation is the number of arrangement of a part or all objects of a set. Here's the formulas for permutation, but we need to define what is factorial notation.

n! is read as "n-factorial" that is:

Definition 1:

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

For instance:

5! = 5(4)(3)(2)(1) = 120

Definition 2:

0! = 1

Now let's have the formula for permutation.

Theorem 1. The number of arrangements of distinct object is n!

Example: In how many ways can you arrange the letters from the word "cat"?

N = n! = 3! = 3(2)(1) = 6

Here's the possible arrangements: cat cta act atc tac tca

Theorem 2. The number of arrangements of n distinct objects taken r at a time is

$_ n {P}_{r} =$n!/(n-r)!

Theorem 3. The number of arrangements of n distinct objects arranged on a circle is:

(n-1)!

On the otherhand, combination is the number of selection of n object taken r at a time that is,

nCr = n!/(n-r)!r!#

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