11-1 Permutations and Combinations

Quick Review

If event M can occur in m ways and event N can occur in n ways, then M followed by N can occur in m · n m middle dot n  ways. The notation n! (n factorial) means n ( n − 1 ) · … · 3 · 2 · 1 . n open n minus 1 close middle dot dot dot dot middle dot 3 middle dot 2 middle dot 1 .  The number of ways to choose r items from a set of n items, without regard to order, is   n C r = n ! r ! ( n − r ) ! . sub n , cap c sub r , equals . fraction n factorial , over r factorial open n minus r close factorial end fraction . .  The number of ways to choose r items from a set of n items and place those items in some order is   n P r = n ! ( n − r ) ! . sub n , cap p sub r , equals . fraction n factorial , over open n minus r close factorial end fraction . .

Example

A vacationer making travel preparations chooses three books from a shelf containing 15 books. How many ways are there to choose three books without regard to order? How many ways are there to choose one book for the trip to the destination, one for the stay, and one for the homeward trip?

Ignoring order, there are   15 C 3 = 15 ! 3 ! 12 ! = 455 sub 15 , cap c sub 3 , equals . fraction 15 factorial , over 3 factorial 12 factorial end fraction . equals 455  ways to choose three books.

There are   15 P 3 = 15 ! 12 ! = 2730 sub 15 , cap p sub 3 , equals . fraction 15 factorial , over 12 factorial end fraction . equals , 2730  to choose three books to read in a particular order.

Exercises

Evaluate each of the following.

5. 3!
6. 9!
7. 4 ! 2 ! fraction 4 factorial , over 2 factorial end fraction
8. 5 ! 2 ! 2 ! fraction 5 factorial , over 2 factorial 2 factorial end fraction
9.   7 C 2 sub 7 , cap c sub 2
10.   4 C 3 +   6 C 5 sub 4 , cap c sub 3 , plus , sub 6 , cap c sub 5
11.   6 P 2 sub 6 , cap p sub 2
12.   4 P 3 +   6 P 5 sub 4 , cap p sub 3 , plus , sub 6 , cap p sub 5
13. Camping On a camping trip, you bring 12 items for 4 dinners. For each dinner, you use 3 items. In how many ways can you choose the 3 items for the first dinner? For the second? For the third? For the fourth?
14. Advertising A newspaper ad includes a telephone number 1-555-DIAL VSW. How many seven-letter arrangements are possible for the phone number using the 26 letters of the alphabet if no letter is used more than once? Express your answer using scientific notation.

11-2 Probability

Quick Review

Experimental probability is based on successes during repeated trials, while theoretical probability is based on number of occurrences in a sample space of equally likely outcomes. When an actual event cannot easily be repeated through numerous trials, a simulation can be used to obtain an experimental probability.

Example

What is the probability that you were born on a Thursday?

Since you had an equal chance of being born on any one of the 7 days of the week, the probability that you were born on a Thursday is 1 7 . 1 seventh , .

Exercises

15. How many possible outcomes are there when a standard number cube is rolled three times?
16. You flipped a coin 70 times and recorded 23 heads. What is the experimental probability of flipping tails?

Find the probability of each event.

17. A rolled number cube comes up 13.
18. A number picked at random from the numbers 1 through 15 is prime.
19. Writing Suppose you have 20 tiles with the numbers 1 through 20. The theoretical probability that a tile chosen at random is a 5 is 1 20 . 1 twentieth , .  If you pick a tile randomly, 20 times, replacing the chosen tile each time, will you get a 5 once? Explain.

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