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## Example Questions

### Example Question #22 : Permutation / Combination

If a team of 7 basketball players win a game and each high-five each other once after a game, how many high-fives take place?

**Possible Answers:**

540

720

5040

21

28

**Correct answer:**

21

To start, the first player makes 6 high-fives. The second player needs to make only 5 new high-fives because he doesn't need to re high-five the first player. The third player needs to only make 4 high-fives and so on.

This adds up to 6 + 5 + 4 + 3 + 2 + 1 = 21

### Example Question #1 : How To Find Permutation Notation

Mr. and Mrs. Jones have invited three other married couples to a dinner party; the eight are to be seated at the table shown above. It is desired that each husband sit directly across from his wife -for example, a husband at Seat 1, his wife at Seat 6.

How many ways can the eight persons be seated to fit this specification?

**Possible Answers:**

**Correct answer:**

There are four pairs of seats directly across from each other: 1-6, 2-5, 3-8, 4-7.

There are four married couples to be placed in these four pairs of seats; since order is relevant, these are permutations. The number of ways to seat these couples is equal to the number of orderings, or permutations, of four elements from a set of four:

,

which is equal to

Within each pairing, there are two ways for the husband to select one seat and the wife to select the other, and there are four such pairings, so, applying the multiplication principle, the number of arrangements is

### Example Question #2 : How To Find Permutation Notation

Mr. and Mrs. Williams have invited three other married couples to a dinner party; the eight are to be seated at the table shown above. Since Mr. and Mrs. Williams need to be close to the kitchen so that they can serve the meal, it is desired that they sit at Seats 1 and 8, although it does not matter which one sits in which seat. There are no other restrictions.

How many ways can the eight persons be seated to fit this specification?

**Possible Answers:**

**Correct answer:**

Since Seats 1 and 8 will be occupied by the Williamses, there are two ways to fill these seats - Mr. Williams in Seat 1 and Mrs. Williams in Seat 8, and the opposite case.

The remaining six persons will be seated in the other six seats; since order is material here, this is a number of permutations. The number of permutations of six elements from a set of six is

,

which is equal to

.

By the multiplication principle, the number of seating arrangements is

.

### Example Question #3 : How To Find Permutation Notation

Yvonne wants to invite seven of her twenty classmates to her birthday party. She and her classmates will all be seated at the above table. Which of the following expressions gives the number of ways she can select seven classmates and seat them, and herself, at the table?

**Possible Answers:**

None of these

**Correct answer:**

One way to look at this is as follows:

Yvonne can sit at any one of the eight seats—eight ways to make this decision.

Yvonne can fill the remaining seven seats with seven out of twenty classmates. Since order is important, this is a permutation of seven elements chosen from a set of twenty; there are ways to make this decision.

By the multiplication principle, there are ways Yvonne can invite seven classmates and seat them and herself at the table.

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