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Permutation sounds complicated, doesn’t it? Here’s an easy way to remember it: And it is. When it comes to permutations, every small detail counts. Alice, Bob, and Charlie are not the same as Charlie, Bob, and Alice (insert the names of your pals here).
Combinations, on the other hand, are relatively straightforward. Details are unimportant. Alice, Bob, and Charlie are identical as Alice, Bob, and Charlie.
Combinations are used for groups while permutations are used for lists (order matters).
A “combination lock” should rather be referred to as a “permutation lock.” It matters how you arrange the numbers.
The hairy details of permutations
Let’s start with permutations, or all of the conceivable outcomes. We’re using the fancy term “permutation,” which means we’ll pay attention to every minute detail, including the sequence in which each item appears. Let’s say there are eight of us:
1: Alice
2: Bob
3: Charlie
4: David
5: Eve
6: Frank
7.George
8.Horatio
How many different ways can we award a first, second, and third place prize to each of the eight contestants? (Platinum, Silver, and Bronze)
medals for graduation examples
We’ll use permutations since the sequence in which these medals are awarded is important. Here’s how it all works:
8 options for the gold medal: A (Isn’t it clever how I made the names match up with letters?) B C D E F G H Assume A wins the Gold.
7 alternatives for the silver medal: B C D E F G H. Assume B is the silver medalist.
Bronze medalists had six options: C D E F G H. Let’s say C takes bronze.
We chose certain people to win, but the specifics don’t matter: we had eight options at first, then seven, and finally six. There were a total of options.
Let’s have a look at the specifics. We had to order three people out of a total of eight. To do this, we began with all alternatives (8) and gradually removed them one by one (7, then 6) until we ran out of medals.
8*7*6=336
The factorial is as follows:
8! = 1*2*3*4*5*6*7*8
The fancy permutation formula is as follows: You have n objects and want to know how many different ways you can order them:
Combinations!
Combinations are simple. Order is irrelevant. It looks the same no matter how you mix it up. Assume I’m a penny-pincher who can’t afford individual Gold, Silver, and Bronze medals. I can only afford to buy empty tin cans.
How many different ways can I give three tin cans to eight people?
In this scenario, the sequence in which we select people is irrelevant. Giving a can to Alice, Bob, and then Charlie is the same as giving a can to Alice, Bob, and then Charlie. In either case, they’re both disappointed.
This brings up an intriguing point: there are some redundancies in this system. Alice Bob Charlie is the same as Alice Bob Charlie. Let’s see how many different ways we can rearrange three people for a moment.
We have three options for the first individual, two for the second, and one for the third. So now we know how to rearrange three persons.
Just a moment… This is starting to resemble a permutation! You duped me!
I certainly did. It’s merely N factorial or N if you have N persons and want to know how many configurations there are for all of them.
So, if we have three tin cans to give away, each option has three! or six variations. To find out how many combinations we have, we simply multiply all of the permutations by all of the redundancies. We have 336 permutations in our example (from above), which we divide by the 6 redundancies for each permutation to get 336/6 = 56.
Conclusion
A combination is a selection of elements from a collection with distinct members in which the order in which they are chosen is irrelevant (unlike permutations).
Given three fruits, such as an apple, an orange, and a pear, there are three possible combinations of two: an apple and a pear, an apple and an orange, and a pear and an orange. A k-combination of a set S is a subset of the set’s k distinct elements.
