Knowing the words for different areas of math isn’t on the SAT, so don’t worry if the word “permutation” sounds to you like something a mad scientist experiments with. Both of the types of questions we’re looking at here have in something in common: you’ll be asked to find the number of possible situation that can be formed by arranging the pieces of some set. The questions below do that, but in different ways.
If a password made of 3 digits only uses the numbers from 1 to 5, inclusive, how many distinct passwords are possible if no digit is used more than once?
If Andy’s breakfast is 3 pieces of fruit taken from a bowl that contains 1 apple,
1 orange, 1 banana, 1 peach, and 1 plum, how many different combinations can his meal consist of?
Although they ask for the same basic thing, we’d go about these questions differently, because order matters in the first question but doesn’t in the second. That is, 234 is not the same password as 432; but an apple, an orange and a banana are the same set of fruit no matter what order they’re eaten in.
Dealing with permutations on the SAT
Permutations are what the first example shows. To answer a question like this, there are two ways to go about it. The first one involves a formula.
k is the number of desired places, which is 3 in the problem, while n refers to the number of different options you are drawing from, which is 5.
While that formula works just fine, it’s not really necessary to memorize it, thankfully. It’s actually easier to use some logic.
Draw boxes for each place in line you have—in this case, there are three numbers in the combination, so we’ll draw 3 boxes.
Each one represents a number in the lock password. There are five possible numbers we could use for the first digit of the password (1-5), so we’ll put a five in the first box.
Because we’ve used one digit, there are only four possible digits that can go in the next box—remember that the question said that no digit is used more than once.
And following the same logic through one more step, we get this.
Now, if you multiply those together you’ll get the answer. That process is probably easier to remember than the formula, so try it out a few times in with different scenarios before you take your SAT.
How SAT combinations are different
Look back at the question about Andy’s breakfast, and consider whether it’s logical and clear that the solution uses the formula below.
No? Good, I’m not alone there. By the way, the r in that formula is the number of members of the group, similar to k in the permutations formula.
And again, rote memorization of the formula will get you there, but there’s a more natural, comfortable way. Let’s just start writing down all of the possible combinations. Each piece of fruit will be assigned a letter: A for apple, B for banana, C for…ummm…orange, D for peach, and E for plum. Alright, so it would have been nicer looking if the question had told us there was a carrot, a dragonfruit, and an entawak (yes, that’s a real thing), but who cares. Using the alphabet is the easiest way to keep from having to write out whole words.
So we start making groups. First write out all the combinations that use A, keeping in mind not to repeat any one letter because he can’t eat the same fruit twice.
ABC, ABD, ABE
Then keep going with B, no longer using A.
Then with C.
And there’s nothing else to use. So we count them all up and see that there are 10 different combinations of fruit Andy might have. A question like this on the SAT generally won’t make you write out more than fifteen different combinations, so it’s actually time-efficient to just do that rather than worrying about the formula.
Just remember to ask yourself, “Does order matter?” If it doesn’t, then start writing out the possibilities. If it does, then draw a box for each place in line.
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About Lucas Fink
Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.
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