2019 worksheet 5 with answers PDF

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Worksheet for 2019 winter quarter which begun January 4, 2019...

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Worksheet 5: Permutations and Combinations If we choose k objects from a set of n objects and put them in a particular order, then we have created a permutation or arrangement. Question: How many arrangements are there of k objects chosen from a set of n? Answer: n( n 1)( n  2) ( n  ( k  1)) P( n, k) Example 1: How many ordered lists of three letters can be made from { A, B, C, D} ? The answer is P(4, 3) 24 . We can list them all as follows: {ABC, ABD, ACB, ACD, ADB, ADC, BAC, BAD, BCA, BCD, BDA, BDC, CAB, CAD, CBA, CBD, CDA, CDB, DAB, DAC, DBA, DBC, DCA, DCB}

If we choose k objects from a set of n objects and don’t care what order they are in, then we have created a combination or subset of the n element set. Question: How many subsets are there of k objects chosen from a set of n? n( n  1)( n  2) ( n  (k  1)) C (n, k )  k! Answer: Example 2: How many sets of three letters can be made from { A, B, C, D} ? The answer is C(4,3) 4 . We can list them all as follows: {{A, B, C}, {A, B, D}, {A,C,D}, {B,C,D}} Note that for each 3 element subset in Example 2, there are six ordered list in Example 1. For instance, the set {A,B,D} corresponds to the six lists {ABD, ADB, BAD, BDA, DAB, DBA} That’s why the answer to Example 1 is 6 times the answer to Example 2. Example 3: Given the digits {1,2,3,4,5,6,7,8,9}, in how many ways can we create a pair of two-digit numbers (x, y) where x uses only odd digits and y uses only even digits. Answer: The number x can be formed in P(5, 2) 20 ways (since order does matter in choosing the two digits) and having done that, the number y can be formed in P(4, 2) 12 ways (again, order matters). Hence, the pair (x,y) can be formed in (20)(12) 240 ways.

Exercises 1. How many 2-element subsets can be formed from the set {A, B, C, D, E}? C (5,2) =10 2. How many different 3-digit numbers having no repeated digits can be made using the digits {2, 3, 5, 7, 8, 9}? P (6,3) = 120 3. How many 5-card hands can be dealt from a standard 52-card poker deck? C (52,5) = 2,598,960 4. How many poker hands are there that contain exactly a 7? (Hint: Form such a hand in two steps: First choose the 7 card. In how many ways can that be done? Then choose the other four cards so that none of them is a 7. In how many ways can that be done? Then multiply.) First there are 4 ways to pic a 7 To pick other 4 cards we need C (48,4) 4* C (48,4) = 4*194,580=778,320 5. If you are dealt a hand at random, how likely is it that the hand contains exactly one 7? C ( 5,1 )∗C (47,3) =0.29947 C (52,4) 6. We have 13 players and we want to assign a player to each of five positions on a basketball team. How many different player assignments can be made? P(13,5) =154,440 7. We have a set of 13 different flowers on a table. How many different sets of 5 flowers can be made? C(13,5) =1287 8. Suppose a class has 11 male and 13 female students. In how many ways could we choose a 4-student subset, if we insist that the subset consist of two male and two female students? C(11,2)*C(13,2) = 4,290 9. Suppose a class has 11 male and 13 female students. In how many ways could we choose two male students to be vice-president and secretary of the class and two female students to be president and treasurer? P(11,2)*P(13,2) = 17,160

Answers 1. 10 2. 120 3. C (52,5) 2598960 Thus, there are just under 2.6 million possible poker hands 4. 778320 5. 0.299 6. 1554440 7. 1287 8. 4290 ways. 9. 17160 ways...