2.2. Permutations and Combinations Compared http://www.ck12.org
In mathematics, we use more precise language:
If the order doesn’t matter, it is a combination.
If the order does matter, it is a permutation.
Say, for example, you are making a salad. You throw in some lettuce, carrots, cucumbers, and green peppers. The
order in which you throw in these vegetables doesn’t really matter. Here we are talking about a combination. For
combinations, you are merely selecting. Say, though, that Jack went to the ATM to get out some money and that he
has to put in his PIN number. Here the order of the digits in the PIN number is quite important. In this case, we are
talking about a permutation. For permutations, you are ordering objects in a specific manner.
TheFundamental Counting Principlestates that if an event can be chosen in pdifferent ways and another
independent event can be chosen inqdifferent ways, the number of different ways the 2 events can occur isp×q. In
other words, the Fundamental Counting Principle tells you how many ways you can arrange items.Permutations
are the number of possible arrangements in an ordered set of objects.
Example A
How many ways can you arrange the letters in the word MATH?
You have 4 letters in the word, and you are going to choose 1 letter at a time. When you choose the first letter, you
have 4 possibilities (’M’, ’A’, ’T’, or ’H’). Your second choice will have 3 possibilities, your third choice will have
2 possibilities, and your last choice will have only 1 possibility.
Therefore, the number of arrangements is: 4× 3 × 2 × 1 =24 possible arrangements.
The notation for a permutation is:nPr,
where:
nis thetotalnumber of objects.
ris the number of objects chosen.
For simplifying calculations, whenn=r, thennPr=n!.
Thefactorial function (!)requires us to multiply a series of descending natural numbers.
Examples:
5!= 5 × 4 × 3 × 2 × 1 = 120
4!= 4 × 3 × 2 × 1 = 24
1!= 1
Note: It is a general rule that 0!=1.
Example B
Solve for 4 P 4.
4 P 4 =^4 ·^3 ·^2 ·^1 =^24
This represents the number of ways to arrange 4 objects that are chosen from a set of 4 different objects.
Example C
Solve for 6 P 3.