Combinations vs. Permutations on the SAT

For many who have learned combinations/permutations there is a tendency to become filled with dread as soon as somebody so much as says the two words. But do not worry – for the SAT the combinations/permutations on the SAT are very basic. For those who have not studied combinations/permutations you are probably wondering, what the heck are those?!? What the heck are permutations? Let’s say you have five chairs and you want to seat five people in those chairs. How many possible ways can you seat them? One way is to count out all the different ways, a process that would take you the entire time of an SAT section. A far more effective way to solve this problem is to use permutations. Whenever you are trying to find how many ways to arrange a group of people/things always use the permutation formula. For the SAT, the permutation formula is really simple: (The number of things)! That last mark (!) is a factorial sign (I’m not that excited about the permutations). A factorial means the following: 5! = 5 x 4 x 3 x 2 x 1 3! = 3 x 2 x 1 10! = 10 x 9 x 8†¦ x 3 x 2 x 1 With the question above, the answer is 5!, because there are five people we are arranging. 5! = 120. What the heck are combinations? Let’s say you are choosing a certain number of people from a larger group of people: The coach can ‘c’hoose 3 players from a group of 6. How many different teams can result? (Notice the ‘c’ in choose cues you to use ‘c’ombinations   not permutations). Unlike permutations, these questions are very rare on the SAT (though permutations aren’t that common either). To solve them you can use the combination formula (which you may have learned in school). I favor the dash method, which follows: ____   ____   ____ Each dash stands for a player. The coach is choosing three players, so there are three dashes. In the first dash, place the number of players the coach can choose from (which is 6). For the second dash, the coach can choose from the remaining players (which is 5). A 4, for the last four players, goes in the last dash. 6  Ã‚  Ã‚  x  Ã‚  Ã‚  5  Ã‚  x  Ã‚  Ã‚  4  Ã‚  =120. Now divide by the number of dashes factorial (3!) = 6. 120/6 = 20. Thus there are 20 different teams that could result. Notice that if I choose Bob, Mike, and Steve as a part of team the arrangement doesn’t matter: Bob, Mike and Steve is the same team as Bob, Steve, and Mike. Here we are just choosing a team; hence we use the combinations formula. If the position were important—one could play center, one forward, the other point guide—we would be dealing with the number of arrangements. Traditionally we would use the permutations formula. But again the dash method makes things much easier: 6  Ã‚  Ã‚  x  Ã‚  Ã‚  5  Ã‚  Ã‚  x   Ã‚  4 Using the dash method and permutations, we do not have to divide by anything. The answer is simply 120. That is if we choose 3 players from 6, and once they are on the team the order matters, we can have 120 unique teams.    Practice Question: Marty has the novels War and Peace, The Sound and the Fury, Crime and Punishment, and Pride and Prejudice on his bookshelf. How many ways can he arrange the books? (A)   6 (B)   8 (C)   12 (D) 24 (E)   42 Answer and Explanation: Take the number of books (4) and stick a factorial sign at the end (4!). Voila, we have our answer: 24. (D). Takeaway As long as you understand the difference between combinations and permutations in this post, you should do a good job on the SAT. To prep for the SAT, do not struggle with really difficult examples that you may find on-line or in math textbooks.