For example, suppose we choose a few items from a large number of different items, and you are trying to figure out how many different ways to do this. You’re trying to find that Number of combinations.

For example, suppose you want to buy 2 cars from a set of 4 different options. Suppose the crowd is {Honda, Toyota, Mercedes, BMW}

How many ways can you choose two cars from this set?

Here are the different ways you can do this.

(Honda, Toyota) (Honda, Mercedes) (Honda, BMW) (Toyota, Mercedes) (Toyota, BMW) (Mercedes, BMW)

Each of the possible choices in this list is called a Combination. Therefore the number of combinations is 6.

It is important to note that the order in which the selections are made is not important with combinations.

For example, the combination (Honda, Mercedes) is the same as (Mercedes, Honda). Both arrangements represent 1 combination.

Combination notation

Combinations indicate the number of ways in which r elements can be selected from n elements.

The notation we use to indicate the number of combinations is C (n, r) or nC.R.

You can also use the following notation. Note that the two notations above and below can be read as “the number of combinations of n selected items r at the same time”.

$$ {n choose r} $$

n is the total number of elements

r is the number of elements selected from the total number of elements

Number of combination formulas

$$ {n choose r} = frac {n!} {r! (nr)!} $$

Let’s review our previous example of choosing 2 cars from a series of 4 choices. Use the formula above to find the number of combinations

In this case n = 4 and r = 2

$$ {n choose r} = {4 choose 2} = frac {4!} {2! (4-2)!} = frac {4!} {2! (2)!} $$

$$ {n choose r} = {4 choose 2} = frac {4 × 3 × 2!} {2! (2)!} = frac {4 × 3} {2!} = frac { 12} {2 × 1!} = Frac {12} {2 × 1} $$

$$ {n choose r} = {4 choose 2} = frac {12} {2} = 6 $$



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