Permutations and Combinations
Permutations and Combinations
If the new SAT is a haunted forest, permutation and combination problems are the deepest, darkest, rarest trees. No, they are the mysterious fluorescent fungus growing on those trees. Permutation and combination problems are almost always hard, and most students skip them because they take so long. But if you’re going for a Math score above 700, you should know how to deal with them. And to deal with permutations and combinations, you first have to know about factorials. If you’re rushed for study time, though, and you’re not trying to score a 700 on the Math section, this would be a good section to skip.
The factorial of a number, represented by n!, is the product of the natural numbers up to and including n:
The factorial of n is the number of ways that the n elements of a group can be ordered. So, if you become a wedding planner and you’re asked how many different ways six people can sit at a table with six chairs, the answer is 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Mutations are genetic defects that result in three-headed fish. A permutation, however, is an ordering of elements. For example, say you’re running for office in California, and there are six different offices to be filled—governor, lieutenant governor, secretary, treasurer, spirit coordinator, and head cheerleader. If there are six candidates running, and the candidates are celebrities who don’t care which office they’re elected to, how many different ways can the California government be composed? Except that California politics are funny, this question is no different from the question about the ordering of six people in six chairs around the table. The answer is 6! = 720 because there are six candidates running for office and there are six job openings.
But, what if a terrible statewide budget crisis caused three California government jobs to be cut? Now only the three offices of governor, lieutenant governor, and spirit coordinator can be filled. The same six candidates are still running. How many different combinations of the six candidates could fill the three positions? Time for permutations.
In general, the permutation, nPr, is the number of subgroups of size r that can be taken from a set with n elements:
For the California election example, you need to find 6P3:
Notice that on permutations questions, calculations become much faster if you cancel out. Instead of multiplying everything out, you could have canceled out the 3 × 2 × 1 in both numerator and denominator, and just multiplied 6 × 5 × 4 = 120.
Permutations and Calculators
Graphing calculators and most scientific calculators have a permutation function, labeled nPr. Though calculators do differ, in most cases, you must enter n, then press the button for permutation, and then enter r. This will calculate a permutation for you, but if n is a large number, the calculator often cannot calculate n!. If this happens to you, don’t give up! Remember, the SAT never deals with huge numbers: Look for ways to cancel out.
A combination is an unordered grouping of a set. An example of a combination scenario in which order doesn’t matter is a hand of cards: a king, an ace, and a five is the same as an ace, a five, and a king.
Combinations are represented as nCr, where unordered subgroups of size r are selected from a set of size n. Because the order of the elements in a given subgroup doesn’t matter, this means that will be less than nPr. Any one combination can be turned into more than one permutation. is calculated as follows:
Here’s an example:
Suppose six people are running for three leadership positions, each of which has the same duties and title. How many ways can this be done?
In this example, the order in which the leaders are assigned to positions doesn’t matter—the leaders aren’t distinguished from one another in any way, unlike in the California government example. This distinction means that the question can be answered with a combination rather than a permutation. So, to figure out how many different groups of three can be taken from a group of six, do this:
There are only 20 different ways to elect three leaders, as opposed to 120 ways when the leadership jobs were differentiated.
Combinations and Calculators
As with permutations, there should be a combination function on your graphing or scientific calculator labeled nCr. Use it the same way you use the permutation key.
That’s it, everything, the whole SAT mathematical banana—from Numbers and Operations to Data, Statistics, and Probability. You’ve now covered every little bit of math that might appear on the SAT. To make your job even easier, at the start of this section is a chart that summarizes the most important SAT math facts, rules, and formulas for quick reference and easy studying.
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