1. Chapter 3b Counting Rules

2. Permutations  How many ways can 5 students in a class of 30 be assigned to the front row in the seating chart?  There are 30 ways to choose the first. For each of the 30, there are 29 choices for the second. For each of these pairs, there are 28 choices for the third, and so on.  Thus there are 30x29x28x27x26 permutations for the first row.

3. Permutations  In general, if there are n objects and you want to select k of them, the number of arrangements that are possible is n P k =n(n-1)(n-2)…(n-k+1).  This deceptively simple idea turns out to be of great importance.  First, note that if you constructed a tree diagram for this problem, this process is like counting the branches.  But, constructing tree diagrams for n>5 or so is not practical.

4. Permutations  Second, note how this is related to conditional probability—the choices in the second step depend on the first, etc.  Although this rule is for counting outcomes, it is easy to see that on a tree diagram we would be labeling the first generation probabilities 1/n, the second generation 1/(n-1), etc.  Thus the probability for any branch is

5. Factorials  Suppose we wanted to find the total number of ways of arranging 5 objects (5 out of 5).  There would be 5x4x3x2x1 ways.  There is a special symbol for this kind of product: n!, called n-factorial, is defined as follows:

6. Factorials  Factorials get huge, fast. 5! =120 10!=3,628,800 20!=2,432,902,008,176,640,000  Thus we can easily get beyond the capabilities of ordinary calculators to deal with these numbers accurately.  It is very helpful to learn a shortcut for dividing factorials.

7. Factorials  Say you had 20!/17!.  Recognize that  Use this idea in reverse to find a new way to write the permutations rule.

8. Combinations  Suppose now that you want to choose 5 people from a class of 30 to be on a committee.  This is called a combination. In contrast to a permutation, the order of objects is not important. In a permutation, we choose one person for each distinct position. In a combination, it is as though we choose them all at once.

9. Combinations  For example: {Bob, Jo, Ann, Judy, Carl} is the same as {Ann, Judy, Carl, Jo, Bob}.  The above example lists two of the permutations of five people. We already know how many permutations there are, total. What we need to do is determine how many permutations are the same, when order is ignored, then divide by that number.

10. Combinations  How many ways are there to order 5 objects? 5! ways. There are 30!/25! permutations of 5 chosen from the class. So there are (30!/25!)/5! combinations.  The symbol for the number of combinations of k items chosen from n is and is read “n choose k” It is also written as n C k (often on calculators). It is defined as follows:

11. Combinations  The formula may look complicated, but it is easy to remember if you think of it this way: The total goes on top, and the denominator splits the total into two groups, number chosen and number not chosen.

12. More on Combinations  Combinations can be used to calculate probabilities for many common problems like card games.  Recall the probability definition that said.  If we can find the number of combinations of cards that fit our definition of a particular “hand,” and divide by the total number of combinations of cards, we can calculate probabilities for “hands.”

13. Poker Hands  First, what is the total number of hands possible in poker? (5-card combinations)  This will be our denominator.

14. Try a Flush  To figure out the numerator, we can think in terms of the choices that have to be made to build the hand.  For a flush, we first have to choose 1 suit out of 4. Given the suit (think tree), we have to choose 5 cards from the 13 in that suit.

15. Not All Flushes  However, we have to make sure we have mutually exclusive definitions. Not all “flushes,” are counted as flushes. Some are straight flushes and royal flushes.

16. Final Results for Flushes  So

17. One Pair  To get one pair (and nothing else) we need to select a number for the pair, then 3 other, different numbers for the remaining cards (order doesn’t matter). However, that is not all. We must also select suits, two for the pair and 1 for each of the other cards.

18. Full House  One more example: To get a full house, you choose one number for the pair, then two suits for it, then another number for the triple, and three suits for it.