Tyler York

June 8, 2023

In this video, Achievable GRE course author Matt Roy explains how to find the possible number of combinations one can make from a limited set of options. This problem type is different from permutations in that the order by which you choose the options does not matter.

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00:00:02 The next problem is a combinations problem, and in this video I'll be describing it to you. The three questions you should ask yourself for any combinations problem. So let's start with reading the question. It says you really like 7 books that you found at the store or at the library. However, you're only able to take home four of them. How many sets of four books could you take home? 00:00:29 So the first of the three questions you should ask yourself is how many decisions do I have to make? In this case, we're going to take home four books, so we have to make four different decisions. Now you should write out four slots here. Next, you should ask yourself how many options do I have per decision for a very first decision. 00:00:56 We get to choose from 7 books, so we have seven options for our next decision. Given that we've only chose we've already chosen one book, we have six options, and so on it goes down 5 to 4. All of these decisions need to be multiplied by each other, and this here would be the answer if the question was how many ways can you arrange 4 out of seven books in line? 00:01:26 But this question asks how many groups of books can you take home, so that's really important. The third question you should always ask yourself is does the order matter? And the order does not matter in this case, because if you were to take home books ABCD, that is the same as if you took home the books DCBA. It does not matter which book gets checked out first or which book you put in your bag first. 00:01:56 That's the same groups of books that you're taking home. In that case, if the order does not matter, then you should do something that I call divide by the number of slots In that case. If the order does not matter, then you should divide by the number of slots factorial. This means you should divide by given that there are four slots. 00:02:25 4 factorial. So you should on the bottom of the fraction, put 4 * 3 * 2 * 1. Now the four on the top and the four on the bottom cancel out. 3 * 2 on the bottom is 6, so that cancels out the six on the top. So we are just left with 7 * 5 on the top. And that's your final answer. There are 35 groups of four books that you could take home. 00:02:54 Out of seven possible books.

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