Data Analysis, Statistics & Probability
Tackling What the #!*@?
Permutation and combination items are the test-makers’ last-ditch effort to bring down your score. The test-takers’ usual reaction to these items is “What the #!*@?” Believe it or not, this visceral reaction can actually help you identify a permutation or combination item. When you find yourself frustrated beyond belief, there is a very good chance you’re looking at one of these items.
Once you’ve collected yourself, follow the three-step method below for solving What the #!*@? items. Not only will it lead you to the correct answer but to a new found feeling of empowerment from shredding the SAT math section.
Step 1: Does order matter? If YES, go to step 2 ONLY. If NO, go to steps 2 and 3.
Step 2: Multiply the factorial of the number of things being arranged to the number of spaces designated for arrangement.
Step 3: Divide by the factorial of the number of spaces in the group.
What the #!*@? in Slow Motion
Now let’s look at the steps more closely, starting with a fairly common What the #!*@? item.
17. Susanne has eleven different medals from her two years of competitive swimming. Unfortunately, the mounting frame she wishes to place them on only has room for two. How many different combinations of medals can Susanne place on her frame?
(A) 21
(B) 33
(C) 55
(D) 66
(E) 110
Step 1: Does order matter? If YES, go to step 2 ONLY. If NO, go to steps 2 and 3.
Susanne is placing two medals onto a mounted frame. Does it matter in which order the medals are placed? No, the word combinations in the item tells us that order does not matter. Recall from our study of the essential concepts that combinations deal with groups of things, while permutations deal with arrangements. In permutations, order matters.
Now let’s move on and complete steps 2 and 3.
Step 2: Multiply the factorial of the number of things being arranged to the number of spaces designated for arrangement.
This step is actually easier than it sounds. How many medals does Susanne have to arrange? 11. How many spaces on her frame does she have for arrangement? 2. So all we do is multiply the first two values in the factorial of 11 (11!):
Susanne has 11 medals to choose from for the first space and 10 to choose from for the second space.
Step 3: Divide by the factorial of the number of spaces in the group.
Because there are two positions on the frame, we go ahead and divide by the factorial of 2, (2!). So now our entire equation looks like this: . That’s it, that’s all! The correct answer is C, 55.
Guided Practice
Try this one on your own.
17. Kevin had a very busy Saturday afternoon. He just bought a new spice rack that holds four spice jars. When he got home, he remembered that he has six different types of spices. How many different arrangements of spices, from left to right, are possible for Kevin’s new spice rack?
(A) 15
(B) 60
(C) 120
(D) 180
(E) 360
Step 1: Does order matter? If YES, go to step 2 ONLY. If NO, go to steps 2 and 3.
Is the item asking about groups or arrangements? If it is a group, then order does not matter. If it is not a group, then it must be an arrangement, and order does matter.
Step 2: Multiply the factorial of the number of things being arranged to the number of spaces designated for arrangement.
Remember your factorials. Find the number of spaces in the arrangement or group.
Step 3: Divide by the factorial of the number of spaces in the group.
How many people or things are in the group? That is the number you’ll use to create a factorial.
Guided Practice: Explanation
Step 1: Does order matter? If YES, go to step 2 ONLY. If NO, go to steps 2 and 3.
The item is asking for arrangements, from left to right. This means that order does matter. Now we go to step 2 only.
Step 2: Multiply the factorial of the number of things being arranged to the number of spaces designated for arrangement.
How many things are being arranged? Six spices. How many spaces on the spice rack are there for them? Four. Now multiply the factorial of the number of spices (6!) down to the number of spaces available. It should look like this:
That’s the number of arrangements, answer E. I hope Kevin doesn’t ask us to help.
Step 3: Divide by the factorial of the number of spaces in the group.
This step isn’t necessary for this item, so move on.
Independent Practice
The explanation for this item is on the following page.
18. The student council has five vacancies since the seniors graduated. Nine talented and aspiring young freshmen have applied to fill the vacancies. How many different groups of freshmen are possible to fill the vacancies?
(A) 60
(B) 120
(C) 126
(D) 7,560
(E) 15,120
Independent Practice: Explanation
Step 1: Does order matter? If YES, go to step 2 ONLY. If NO, go to steps 2 and 3.
The key word in this item is groups, which tells us that order does not matter. On to steps 2 and 3.
Step 2: Multiply the factorial of the number of things being arranged to the number of spaces designated for arrangement.
How many people or things are being arranged? Nine freshmen. How many spaces are there on the student council for them? Five. The factorial of 9, (9!), multiplied down five spaces looks like this:
Step 3: Divide by the factorial of the number of spaces in the group.
How many freshmen are going to be in the group? Five. So we are going to divide by the factorial of 5, (5!). Here’s our entire equation:
That’s it! There is the possibility of 126 groups of 5 freshmen to fill the vacancies of the student council. Our answer is choice C, 126.
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