Combinations

Definition of Combinations

A selection of some or all of a number of different objects. It is an un-ordered collection of unique sizes. The order of occurrence if the objects is not important.

Formula of Combinations

Example

To simplify this, consider that :

That neat little trick allow us to enormously simplify the combinations formula :

Keywords of Combinations

• Selection

Types of Combinations

Combinations with Repetition

Let us say there are five flavors of ice cream : banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let’s use letters for the flavors : {b, c, l, s, y}.

Example selections include

• {c, c, c} (3 scoops of chocolate)
• {b, l, v} (one each of banana, lemon and vanilla)
• {b, v, v} (one of banana, two of vanilla)

Now, I can’t describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.

So it is like we are ordering a robot to get our ice cream, but it doesn’t change anything, we still get what we want.

We can write this down as (arrow means move, circle means scoop).

In fact the three examples above can be written like this :

{c, c, c} (3 scoops of chocolate):
{b, l, v} (one each of banana, lemon and vanilla):
{b, v, v} (one of banana, two of vanilla):

OK, so instead of worrying about different flavors, we have a simpler question : 
“How many different ways can we arrange arrows and circles?”

Formula of Combinations in Repetition

Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

The easiest way to explain it is to :

• Assume that he order does matter (i.e. Permutations),
• Then alter it so the order does not matter.

But many of those are the same to us now, because we don’t care what order!

Example

Let us say balls 1, 2 and 3 are chosen.

These are the possibilities :

Order does matter	Order doesn't matter
1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1	1 2 3

So, the permutations will have 6 times as many possibilities.

In fact there is an easy way to work out how many ways “1 2 3” could be placed in order, and we have already talked about it.

The answer is :

So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren’t interested in their order any more) :

Formula of Combinations without Repetition

That formula is so important it is often just written in big parentheses like this :

And is also known as the Binomial Coefficient.

Notation

As well as the “big parentheses”, people also use these notations :

Example

So, our pool ball example (now without order) is :

Or we could do it this way :

It is interesting to also note how this formula is nice and symmetrical :

In other words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations.

Pascal’s Triangle

Here is an extract showing row 16 :

Here are the examples with solutions :

Example 1

Rohan has 3 shirts and 2 pants, in how many are the combinations possible?

He can select any shirt from 3 shirts and any pant from 3 pants.

Example 2

Notice how the cancellation occurs, leaving only 2 of the factorial terms in the numerator. A pattern is emerging when finding a combination such as the one seen in this problem, the second value (2) will tell you how many of the factorial terms to use in the numerator, and the denominator will simply be the factorial of the second value (2).

Example 3

There are 12 boys and 14 girls in Mrs. Schultzkie’s math class. Find the number of ways Mrs. Schultzkie can select a team of 3 students from the class to work on a group project. The team is to consist of 1 girl and 2 boys.

Order, or position is not important. Using the multiplication counting principle,

References :

• https://www.mathsisfun.com/combinatorics/combinations-permutations.html
• http://magoosh.com/gmat/2012/gmat-permutations-and-combinations/