Combinations

When we dealt with permutations, order was important. But when dealing with combinations, order is not important. We just want to see if the object(s) have been selected, not in a particular order.

Let's say that we are dealing with the letters a, b, c, d. Here are the combinations of 3 letters:

a b c
a b d
a c d
b c d

This leads to the form

n C r

where n is the number of objects you have and r is the number you are selecting. In a more general sense, combinations are a subset of permutations. So another way of thinking can be the number of permutations of the objects divided by the number of combinations you want from them:

So using the above example, we get the following:

which is indeed correct as we observe there were 4 combinations of the letters a b c d.

(a)
How many combinations of 10 things be taken 6 at a time?

(b)
How may ways can you select a committee of 4 students 
from a class of 12 students?

(c)
Evaluate 5 C 3

(d)
Evaluate 7 C 7

(e)
Evaluate 8 C 0

Solutions:

(a)

(b)

(c)

(d)

(e)

Repeated Combinations

Sometimes, you want to deal with objects where you may need to use them more than once. One such example is the repeated combinations of the letters a, b and c:

a a a
a a b
a a c
a b b
a b c
a c c
b b b
b b c
b c c
c c c

There are a total of 10 repeated combinations in the above. This is a slightly modified formula than the usual:

A problem will usually state whether or not there is repetition involved with the combination.

A good example to think of is an ice cream parlor. Say you are given a selection of 3 scoops of ice cream from 5 flavors. It is acceptable to have 3 scoops of the same kind or 2 scoops of 1 kind and 1 of another etc...The formula takes this into account.

(a)  
Find 6 C 2 with repeated combinations.

(b)  
Find 7 C 7 with repeated combinations

(c)
Find 16 C 0 with repeated combinations

Solutions:

(a)

(b)

(c)

Lottery Jackpots

A practical example of combinations is any lottery jackpot game. We will examine just why you should NOT play the lottery!!!

Let's take the New York State lottery for example. This lottery system uses 59 white balls with 6 of them being drawn to win. Since the order of the numbers does not matter, we will use the combinations formula:

That means there are over 45 million different combinations of numbers! Yikes indeed! That also means the odds of winning are 1 in 45,057,474 or less than a 0.0000002% chance of winning the lottery!!!

Another example is the Mega Millions lottery system. This time, they have 2 parts to it; the white ball section with 56 white balls where you need to match 5 and the mega ball section featuring 46 numbers where you need just 1 to win. So this are two combinations that are multiplied together to get the outcome of the game:

White Ball section:

Mega Ball section:

46 C 1 = 46

And now you need to multiply the number of white ball combinations with the number of mega ball combinations since each of the white ball combinations is paired with an orange mega ball:

W: 1 2 3 4 5   M: 1
W: 1 2 3 4 5   M: 2
W: 1 2 3 4 5   M: 3
W: 1 2 3 4 5   M: 4
W: 1 2 3 4 5   M: 5
W: 1 2 3 4 5   M: 6
...
...
W: 1 2 3 4 5   M: 46

The final result is: 3,819,816 x 46 = 175,711,536

So the odds here are less than a 0.000000005% chance of winning the lottery with this system!!!! Here's some friendly advice, DON'T PLAY IT!!!!

Challenges

After viewing this tutorial, you should be able to solve the following challenges:

Probability 3
Probability 4
Probability 7
Probability 8
Probability 9

Good luck! Remember to register for a FREE account today to submit the answers to the challenges!