SAT Math : Permutation / Combination
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Example Questions
Example Question #21 : Permutation / Combination
There are seven unique placemats around a circular table. How many different orders of placemats are possible?
Since the table is circular, you need to find the total number of orders and divide this number by 7.
The total number of different orders that the placemats could be set in is 7! (7 factorial).
7!/7 = 6! = 720
Note that had this been a linear, and not circular, arrangement there would be no need to divide by 7. But in a circular arrangement there are no "ends" so you must divide by N! by N to account for the circular arrangement.
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?
28
720
5040
540
21
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 #852 : Arithmetic
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?
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 #852 : Arithmetic
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?
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 #853 : Arithmetic
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?
None of these
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 
By the multiplication principle, there are 
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