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Unformatted text preview: CHAPTER 1: Basic Principle of Counting – r experiments to be performed, each with n possible outcomes, total of n 1 x n 2 x…x n r total outcomes. General formula for permutations (order matters so abc ≠ bca: Given n objects, there are n! different ways to arrange them. There are (n-1)! Ways to arrange the n elements in a circle. ! ! ! ! 2 1 r n n n n different permutations of n objects, of which n1 are alike, n2 are alike, etc. ! )! ( ! r r n n r n- = = ! ) ( r n r is the number of possible combinations of size r that could be made from n objects when order is irrelevant. The numerator of the last term is just the notation for n PERMUTE r. Each expression is known as a binomial coefficient. - + -- = r n r n r n 1 1 1 Binomial Theorem: ∑ =- = + n k k n k n y x k n y x ) ( If there are n distinct objects that we want to divide into r different groups of size n1,n2,…nr, we denote it: ! ! ! ! 2 1 ,... 2 1 r r n n n n n n n n = Balls into boxes equations (or elevators): 0^0^0^0^…^0, n objects denoted by 0, and choose r-1 ways to put in a barrier ^ -- 1 1 r n distinct positive integer-valued vectors (x1,x2,..xr) satisfying x1+x2…+xr = n, x>0 If each xr just needs to be non-negative (urns can be empty, not every floor needs someone to get off at it) then the equation goes to: -- + 1 1 r r n Chapter 1 Example Problems: College planning committee of 3 freshmen, 4 sophomores, 5 juniors and 2 seniors. How many subcommittees are possible if 1 person is chosen from each class? Answer: 3*4*5*2 = 120. Similar problems: Total of different ways to order dinner at a restaurant give-rn appetizers, entrees, desserts and drinks. How many 7-place license plates with first 3 as letters and last 4 as numbers? Answer: 26*26*26*10*10*10*10 = 175,760,000. What if no replacement is allowed on license plate problem? Answer: 26 *25*24*10*9*8*7 = 78,624,000 Similar problems: Possible phone numbers How many different batting orders possible for 9 players? Answer: 9! Have 4 math books, 3 chem, 2 history and 1 language. If you want to keep same subjects together, how many ways are there to order the books on the shelf? Answer: 4! Ways to order math books * 3! Chem. * 2! History *1! Language, but there are also 4! To arrange the subjects before arranging the books within the subject, so total ways is 4!4!3!2!1!=6912 How many different ways can you arrange the letters P E P PE R? Answer: 6! Ways to arrange 6 letters, but with 3 P’s and 2 E’s, many of the arrangements will look the same, so divide by different ways to arrange those letters. 6!/(3!2!) = 60 Given 5 women and 7 men, how many committees of 2 women and 3 men can be formed? Answer: (5 choose 2)*(7 choose 3) = 350. What if two of the men are fighting? men are fighting?...
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- Fall '06
- Probability, Probability theory, ways, ACES