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The December version of the ACT contained more “high-level” math problems. We believe this is a trend we are likely to see in future ACT exams, therefore we are adjusting our curriculum. 

 

What’s an example of a high-level math problem?

Combinations and Permutations! (Huh?)

Math problems that ask students to distinguish between combinations and permutations can be perplexing and intimidating. Below we walk through a couple examples to help explain the difference.

What is a Permutation?

In a permutation, the order of the items DOES matter. Permutations (nPr) are used for lists, or a grouping when the order of each item is important, such as a lottery number or a lock. In a permutation function, “n” refers to total possible options and “r” refers to the number of items that are selected.

Example:

In the “Small Dogs Go to Heaven” dog competition, a blue ribbon will be awarded to the loudest dog, a red ribbon to the second loudest dog, and a yellow ribbon to the third loudest dog.  There are 6 dogs in the competition: Rover, Spot, Harry, Lucy, Stella, and Dotty. How many different ways can the ribbons be awarded?

This is a permutation question. The order in which they are arranged makes a difference. If we say Rover, Lucy, and Dotty won, we would assume based on the order that Rover got the blue ribbon, Lucy got the red, and Dotty got the yellow. But another possible arrangement or “permutation” of this grouping could be Lucy, Dotty, and Rover, and the outcome of who received which award would change.

The formula looks like this:

\displaystyle{P(n,k) = \frac{n!}{(n-k)!}}

Want to do that with a calculator?

If you are using the TI-83 or Ti-84:

  1. Enter the “n” value.
  2. Press the MATH button.
  3. Scroll over to PRB.
  4. Choose nPr.
  5. Enter the “r” value.

What is a Combination?

In a combination of items, the order does NOT matter.  Combinations (nCr) are perfect for counting clusters, groups, teams, and committees. Similar to a permutation function, “n” refers to total possible options and “r” refers to the number of items that are selected. 

Example:

Olivia wants to take 3 of her dogs on a morning walk. If Olivia owns 6 dogs (Rover, Spot, Harry, Lucy, Stella, and Dotty), how many different groups of dogs can she choose for her walk?

This is a combination question because once Olivia chooses three dogs to bring, the order of the dogs doesn’t matter.  If she brings Spot, Lucy, and Stella OR Stella, Spot, and Lucy, it’s still the same group of dogs going on the walk.

The formula looks like this:

\displaystyle{C(n,k) = \frac{n!}{(n-k)!k!}}

Want to do that with a calculator?

If you are using the TI-83 or Ti-84:

  1. Enter the “n” value.
  2. Press the MATH button.
  3. Scroll over to PRB.
  4. Choose nCr.
  5. Enter the “r” value.
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six dogs

Combinations & Permutations

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