Permutations vs. Combinations: What's the Difference?

In summary, permutations are all possible arrangements with the sequence of the objects considered. When the sequence of objects is not considered, then it results in a combination. Combinations are all possible arrangements of the objects with themselves, not with other objects.
  • #1
DecayProduct
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Homework Statement



Not really a problem, just my understanding.

What is the difference between them? I know the formulae are different. They seem to be the same thing, that is, n objects taken r at a time. Any help in clarifying this would be appreciated.
 
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  • #2
Permutations are all possible arrangements with the sequence of the objects considered. When the sequence of objects is not considered, then it results in a combination.
 
  • #3
Do you know about vectors and sets? Suppose you have a set of 3 objects, S = {1, 2, 3}. Permutations of 2 objects count the number of 2-object vectors you can construct from this set (without replacement), so you have the size of the set of vectors {(1, 2), (2, 1), (1, 3), (3, 1), (2,3), (3, 2)}, which is 6 distinct pairs. The amount of combinations of two objects formed from objects of this set is the size of the set of sets {{1, 2}, {1, 3}, {2, 3}} which is 3 distinct pairs. You count combinations when you are only concerned about which objects are in each bag or string of text for example, not the order in which they are in the bag. Permutations matter when you are concerned about the order in which the objects are placed in each bag or text string.
Ie., if you are counting 3-letter words, combination counting would count the words DOT and TOD as the same word, while permutation counting notes that the letters are in a different order and counts them as different words.
 
  • #4
Suppose you have a club of 50 people and you want to choose a 3 people as "president", "vice president", and "secretary". You can choose any of the 50 people as president, then any of the 49 remaining people as vice president, then any of the 48 remaining people as secretary. There are, then 50*49*48 ways of doing that. One way to write that is
[tex]50*49*48= \frac{50*49*48*\cdot\cdot\cdot*3*2*1}{47*46*\cdot\cdot\cdot*3*2*1}= \frac{50!}{47!}= \frac{50!}{(50-3)!}[/tex]. This is a "permutation" problem because
"Bill for president, Mary for vice president, Charles for secretary" as different from "Mary for president, Charles for vice president, Bill for secretary".

If instead you want a three person "operating committee" then it doesn't matter which is chosen first, which second, or which third. Now we do not want to treat "Bill for president, Mary for vice president, Charles for secretary" as different from "Mary for president, Charles for vice president, Bill for secretary". Now, since order in not important, it is a "combinations" problem. Since there are 3! ways of writing any three things, to discount interchanging the same three people, we need to divide by 3!:
[tex]\frac{50!}{47! 3!}= _{50}C_{3}[/tex].
 
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  • #5
Halls I think you meant to put [tex] \frac{50!}{47!}[/tex]
 
  • #6
You guys are great. I completely get it now!
 
  • #7
NoMoreExams said:
Halls I think you meant to put [tex] \frac{50!}{47!}[/tex]
Thanks. I had written "\fra " instead of "\frac"! I haved editted it now.
 

FAQ: Permutations vs. Combinations: What's the Difference?

What is the difference between permutations and combinations?

Permutations and combinations are both ways of selecting items from a larger set. The main difference is that permutations take into account the order in which the items are selected, while combinations do not. For example, if you have 3 letters A, B, and C, then the permutations would be ABC, ACB, BAC, BCA, CAB, and CBA, while the combinations would only be ABC, ACB, and BCA.

How do I calculate the number of permutations?

The number of permutations can be calculated using the formula n!/(n-r)!, where n is the total number of items and r is the number of items selected. For example, if you have 5 letters and you want to select 3 of them, the number of permutations would be 5!/(5-3)! = 5!/2! = 60.

What is the difference between permutations with repetition and without repetition?

Permutations with repetition occur when you are selecting items from a set with repeated elements, meaning you can select the same item more than once. For example, if you have 3 letters A, B, and C and you want to select 2 of them, the permutations with repetition would include AAB, AAC, BBA, BBC, CCA, and CCB. Permutations without repetition occur when you are selecting items from a set without repeated elements, meaning each item can only be selected once. Using the same example, the permutations without repetition would only include AB, AC, BA, BC, CA, and CB.

What is the difference between combinations with repetition and without repetition?

Combinations with repetition occur when you are selecting items from a set with repeated elements and the order of selection does not matter. For example, if you have 3 letters A, B, and C and you want to select 2 of them, the combinations with repetition would include AA, AB, AC, BB, BC, and CC. Combinations without repetition occur when you are selecting items from a set without repeated elements and the order of selection does not matter. Using the same example, the combinations without repetition would only include AB, AC, and BC.

How do I calculate the number of combinations?

The number of combinations can be calculated using the formula n!/r!(n-r)!, where n is the total number of items and r is the number of items selected. For example, if you have 5 letters and you want to select 3 of them, the number of combinations would be 5!/3!(5-3)! = 5!/3!2! = 10.

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