COT3100 Summer 2001 Recitation 05 10 Combinatorics 1 Consider a set of five distinct computer science books three distinct math books and two distinct art books a In how many ways can these books be arranged on a shelf 10 b In how many ways can these books be arranged on a shelf if all five computer science books are on the left and both art books are on the right 5 3 2 In how many ways can these books be arranged on a shelf if all books of the same discipline are grouped together 5 3 2 3 c In how many ways can these books be arranged on a shelf if the two art books are not together There are 9 arrangements where two art books are together Then there are 10 2 9 arrangements where two art books are not together 9 arrangements for each of two permutations of two art books Ans 8 9 2 a How many integers should be selected from the first 10 positive integers 1 2 10 to ensure that there exists a pair of these integers with the sum equal 11 6 b How many integers should be selected from the first 10 positive integers to ensure that there exist at least two pairs of these integers with the sum equal 11 7 3 A club has 25 members a In how many ways the four members can be chosen to serve on an executive committee C 25 4 25 21 4 b How many ways are there to choose a president vice president secretary and treasurer P 25 4 25 24 23 22 4 How many strings of six lowercase letters from the English alphabet contain a the letter a at least one and any other letters may be used any number of times 266 256 b the letters a and b in consecutive positions with a preceding b with all the letters distinct 24 23 22 21 5 You can represent the result as a two step task First you select four letters in addition to a and b Since each letter can be used only once you have 24 23 22 21 ways to perform this The second step is to insert an ab tag You have 5 slots to do this By the product rule the final answer is the product of these two numbers d the letters a and b where a is somewhere to the left of b in the string with all letters distinct 24 23 22 21 15 1 Let s consider a two step task again First arrange four letters out of 24 and this step can be performed in 24 23 22 21 different ways The second step is to insert a and b in 5 slots between four other letters Since a must come to the left of b you perform this task in 5 4 3 2 1 different ways By the product rule the answer is 24 23 22 21 15 5 Suppose a 5 card poker hand is drawn from the 52 card deck a How many hands contain three cards of one kind and two cards of a second kind 156 4 6 3744 Consider any outcome as the three step procedure First there are P 13 2 13 12 156 ways to select two different kinds out of 13 the kind of the three cards and the kind of two cards After this you have C 4 3 4 choices for the suites of 3 cards of one kind And finally suites for the two cards of the same kind can be selected in C 4 2 6 different ways The answer is 156 4 6 3744 b How many hands contain all five cards of different kind any suit 1287 45 1 317 888 Let s first select five different kinds of cards out of 13 available This task can be done in C 13 5 13 8 5 1287 ways Then we have four choices for the suit of each card i e there are 45 choices to perform the second step The answer is 1287 45 1 317 888 c A flush is a hand when all cards have the same suit How many five card flushes are possible Ans 4 13 12 11 10 9 5 5 148 C 13 5 ways to choose 5 cards from 13 with the same suit It should be multiplied by 4 different ways to choose a suit d A straight is a hand with five consecutive cards How many five card straights are there 9 45 There are 9 ways to select five consecutive cards Each card can have four suits so there are 45 choices for suits e How many 5 card straight flushes are there 36 4 choices for a suit and 9 choices for the 5 consecutive cards from 13 2
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