# Statistics and Combinations

Combinations:

Selecting only some or all the things form a group of given choices,things,people,ect, is called combination.

Combination is used when only selection is the criteria and the arrangement is not taken into account. Suppose we want to know the number of different ways of selecting a queen card from a pack of 52 playing cards, then we use the method of combination, since the order of selection is not important here.

Suppose there are n things and we want to select only r out of them, we use the combination method represented by nCr read as n combination r.

The formula is given by

Let us look at a simple example to understand the concept of combinations.

Suppose there are 7 boys in a class. I want to select 5 boys only for the school cricket team. In how many different ways can I do the selection?

First we have to read the question carefully and try to understand if the order of selection is important. As we understand form the given question, the order does not hold any importance; all I want is just to select 5 boys from the 7 available. Hence we use the combination formula here to find in how many ways I can do the selection in.

Here n=7 and r=5

Hence

=21

Hence there are 21 different ways in which I can select the 5 boys. Here when we speak about the different ways, it means the different combinations of selection of boys.

For instance consider the 7 boys

1,2,3,4,5,6,7

One way of selection is by selecting the first boys boys,i.e; 1,2,3,4,5

Another way of selection is 1,3,5,7,6

Like this we have 21 different ways of selecting the boys.

The above discussed example is a simple one. Let us extend the same example to have a better understanding of the concept.

Suppose in the same class there are 6 girls. Now I want to form a new committee of 5 members. For this sake I want to select only two boys and 3 girls. In how many different ways can this be done?

Here we have two groups of people namely boys and girls. Form each of these I should select 2 and 3 students respectively. This can be done in the following manner:

• First 2 boys from the 7 available. This can be done in 7C2 ways

• Then select 3 girls from the 6 available. This can be done in 6C3 ways.

• Now to combine both, we multiply 7C2*6C3.
In the above manner if I do the selection, sometime no girl may be selected into the committee and sometimes no boy may be selected in to committee. To avoid such a situation I put one more restriction stating that at least one boy must be included in the committee.

Now the different choices for us are:

1 boy and 4 girls- this can be done in 7C1*6C4 ways

2 boys and 3 girls- this can be done in 7C2*6C3 ways

3 boys and 2 girls-this can be done in 7C3*6C2 ways

4 boys and 1 girl-this can be done in 7C4*6C1 ways

5 boys and no girls-this can be done in 7C5 ways

Now to find the total number of ways in which this selection can be done, we add the three different choices.

7C1*6C4+7C2*6C3 +7C3*6C2+7C4*6C1+7C5

Lets us look into one more example which is more practical in nature.

An advertiser has two strategies A and B for advertising one of his new product. Strategy A has 5 different choices and strategy B has 4 different choices. He can use any 4 of the available choices by selecting at least one form strategies A and B.in how many ways can he do this?

Solution : the advertiser has 5 choices in strategy A and 4 choices in strategy B. he has to select a total of 4 choices by selecting at least one from both the strategies.

The different ways in which we can do this is

• choice form strategy A and 3 choices form strategy B- this can be done in 5C1*4C3 ways

• choices from strategy A and 2 choices form strategy B- this can be done in 5C2*4C2 ways

• choices from strategy A and 1 choice from strategy B- this can be done in 5C3*4C1 ways
Totally the different number of choices are 5C1*4C3+5C2*4C2+5C3*4C1.

As illustrated above the applications of combinations is many. Further when we learn the concept of probability, we extensively use the concept of combination.

When there can be no repetitions of the available choices cannot be done, we always use the concept of combinations. It has a lot of practical applications and plays as a basis for probability theory.

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