A Guide to the GRE/Arrangements

Arrangements
A permutation is an arrangement where the order is important. In this case, the number of possible arrangements of elements is (possibilities for spot 1)(possibilities for spot 2)... This can be represented by a formula as follows:

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


 * $$P(n,k)$$ is notation expressing a number of permutations of n items, taking k items at a time.
 * $$x!$$, read as 'x factorial', signifies the product of all the natural numbers from 1 to $$x$$. For example, 10! = 1(2)(3)(4)(5)(6)(7)(8)(9)(10).

A combination is an arrangement where the order does not matter. The combination formula is the same as the permutation formula, except everything is divided by $$k!$$, i.e. the number of spots factorial:

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

Suppose Kathryn is packing tie-dye shirts to take on vacation, and will pick 3 out of her 27 tie-dye shirts to take with her. How many assortments of shirts can she bring? Order is not important in this question - selecting shirt 3, shirt 5, and shirt 7 is the same as picking shirt 7, shirt 3, and shirt 5. Thus the number is:

$$C(n,k)=C(27,3)={\frac{27!}{24!3!}}={\frac{27!}{6(24!)}}={\frac{27 \cdot 26 \cdot 25}{6}}=2,925$$

Thus, there are 2,925 possible shirt arrangements.

If order was important above - suppose Kathryn was picking a specific shirt for the first, second, and third days - the formula would not be divided by 3!. It would simply be (spot 1)(spot 2)(spot 3), or (27)(26)(25), and would yield 17,550 possibilities.

Note that the number of possibilities descends (27, 26, 25) because the shirts can't be reused. If Kathryn was picking tie-dye shirts to wear on the next three Mondays, and had an opportunity to wash and reuse the same one, the formula would be (27)(27)(27).

Practice
1. A pizza parlor has 8 different toppings available. How many different pizzas with 3 toppings can be made?

2. Lee has 4 different colors of paint to use on his model airplane. He will paint the body one color, the tail rudder a different color, and the nose a different color yet. How many different ways could Lee paint his airplane?

3. A soup recipe calls for any 3 of a list of 6 herbs. How many different groups of 3 herbs can be put in the soup?

Answers to Practice Questions
1. 56

On any arrangement question, ask two questions - is order important, and, can the elements be reused?

Order is not important - pepperoni, sausage and olives is the same as olives, sausage and pepperoni. The elements can not be reused - a pizza can't have sausage, sausage, and sausage as its toppings. It either has sausage or it doesn't. Thus, the formula is:

== 56

2. 64

Order is important - a red nose and a blue rudder is different than a blue nose and a red rudder. The colors can be reused - each of four is available. The formula is thus:

4(4)(4) = 64

3. A soup recipe calls for any 3 of a list of 6 herbs. How many different groups of 3 herbs can be put in the soup?

Order is not important - thyme, parsley, and oregano is the same as oregano, parsley, and thyme in the soup. The elements cannot be reused - the soup must have 3 of the 6 herbs. The formula is thus:

== 20