Combinations can be associated with permutations. The difference is that in combinations we do not care about the order as opposed to permutations where order does matter.
Combinations vs permutations
How many different 4-person groups can we make out of 15 persons? A different way of formulating the same question could be: How many combinations of 15 persons can be made for selecting 4 at a time? We only care about who are selected for the teams. Order does not matter.
The question above is a combination question as opposed to if we had been asked to find the number of different arrangements the teams could be made considering the different positions that each person could have on the team. This would have been a permutation question.
Doing combination, we only care for the number of different groups of people and not where they are positioned.
Let’s see the the difference between combination and permutation based on the question above:
Question 1: How many different 4-person groups can you make out of 15 persons when order does not matter? This is combination.
Question 2: In how many different ways can the 4-person groups be arranged when order matters? This means that the same 4 persons can have different positions in the group and as such these same four persons can be arranged in 4! (=4x3x2x1(=24)) different ways. This is permutation.
Calculation for Question 1 (combination):
1,365 different 4-person groups can be made from 15 persons.
Calculation for Question 2 (permutation):
The 15 persons can be arranged in 32,760 different 4 person groups when each person can take each of the four positions in the group.
So, a group of 15 persons can be combined into 1,365 different 4-persons groups (combination) and be arranged in 32,760 differently arranged 4-persons groups where one person can take all four positions (order matters => permutation).
Let’s run a few more combination examples:
Example A: How many different 11 players groups can be made out of a total of 18 players?
31,824 different groups.
Example B: The sales department is composed by 8 women and 6 men and the company aims to create a project group of 2 women and 2 men. How many different project groups can be made?
In this example we will multiply the possible number of combinations from the women’s group with ones of the men’s group. So, we will say 8 Choose 2 times 6 Choose 2 => 8C2 x 6C2:
420 different project groups can be made that are composed by 2 women and 2 men.
Example C: Say you have 4 paid vacation days which I must use during the current year and that there are 30 labor days left in the year. In how many ways can you use your paid vacation days? This is a 30 Choose 5 combinatoric:
So, you have 27,405 different ways that you can use those last 5 vacation days left in the year.
Combination with MS Excel
The =COMBIN formula is combination in Excel:
Learnings on combination
- Khan Academy page overview for videos on probability and combinatorics
- Khan Academy video: Intro to combinations
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