Another Probability Video – What are the Odds?
Probability – Permutation and Combination
Hey guys. Welcome to this video on Probability.
In this video we will take a look at two different types of probability using permutation and combination.
Now, the first thing you need to know about permutation and combination is when to use them.
When working with a problem where permutation or combination is needed, to distinguish which one all you need to do is ask yourself the question “does order matter?”
For example, if I tell you that I made a quiche with sweet potatoes, broccoli, and bell peppers it doesn’t matter which order I say it in. I could say I made a quiche with broccoli, bell peppers, and sweet potatoes and it would matter. So, in this case order is not important. However, if I tell you the password to my computer is 876, then the order of those numbers is important. If you tried 786, you would be denied access.
Now, when order is not important, like in the first example, then it is a combination; but, when order is important it is a permutation.
There is a little joke that people often make… they say “a combination lock should really be a permutation lock” (in old man voice).
Alright, let’s take a look at a couple different problems. We will look at the formula for both permutation and combination, as well as how to spot whether or not order is important.
An art gallery has twelve paintings by a local artist and wants to arrange four of them on the same wall. How many ways are there to do this?
Alright, let’s look at our problem and identify whether or not order is important. Well, if the art gallery wants to arrange them a specific way then the order must be important. So, basically what we need to find is how many different ways in which they can be ordered.
Now, the formula for permutation is:
N_P_r= N!/(N-r)! N= is the total # in the set
r= how many elements are in the arrangements
So, if N is the total number in our set then N is equal to 12, and r is equal to 4.
12_P_4= 12!/(12-4)! = (12*11*10*9*8*7*6*5*4*3*2*1)/ (8*7*6*5*4*3*2*1)
Once, we write this out, we can see that we can cancel out our 8 through one from the top and the bottom, and we are left with:
Another way to think through what is going on is this:
You have four different places on a wall, so that is 1 , 2, 3, 4. Now, in the first place on the wall there are 12 different painting options to choose from, and once we’ve chosen one there will be 11 options leftover to go in the second spot on the wall, then once we choose one from the 11 there are 10 painting options left to go in the third spot on the wall. Then, lastly, once we choose from the 10 there are 9 paintings left to go on the wall.
So, the total permutations, the different ways that we could arrange these 4 paintings, since we care about the order in which we place them on the wall, that is 12 times 11 times 10 times 9.
Again, we can see that this is the same thing as (12x11x10x9x8x7x6x5x4x3x2x1)/(8x7x6x5x4x3x2x1) which is equal to 12!/8!.
Well, how did we get 8!? Well, recall our original permutation formula. This 12!/8! Is the same thing as 12!/(12!-4!) which is how we find out how many permutations 12 things can place themselves into 4 different positions. Now, this 12!/(12!-4!) is our original permutation formula. Where we have nPr = n!/n!-r!. Each of these different ways are equal, my goal in showing you each is just to help you get down to the foundation of where we got this permutation formula.
Let’s take a look at an example of a combination problem.
A person playing poker is dealt 5 cards. How many different hands are possible for the poker player to have been dealt.
Now, for this problem we don’t care about the order in which the cards are placed in this person’s hand. Order doesn’t matter for this problem, so we know that we will need to use combination.
Here is our combination formula:
N_C_r = N!(r!(N!-r)!
N is our total number of cards. In a standard deck of playing cards there are 52 cards, so N=52. r is the number of cards in the person’s hand at one time, so r=5.
We have to be careful, because the formula for permutation and combination are very similar. So, let’s talk about the only difference between the two, and what that difference is representing.
The difference is that in our combination formula we have an r! Being multiplied by our N!-r!, but we don’t have that in our permutation formula. So, what does that mean? What does this r! Being multiplied by everything in the denominator tell us? It tells us that we are adjusting our permutation formula by reducing it by how many ways the cards could be in order, to get our combination formula. The reason we reduce it by how many ways the cards could be in order is because we don’t care about the order. That is the very reason we are using the combination formula, because order does not matter.
Hopefully, that helps you to better understand the fundamental difference between permutation and combination.
Now, let’s plug in our number to our combination formula, and solve.
52_C_5 = 52!(5!(52-5)! = 52!/5!*47! = 2,598,960
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See you guys next time!