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MIT6_042JS10_lec28_sol

MIT6_042JS10_lec28_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science April 14 Prof. Albert R. Meyer revised April 12, 2010, 700 minutes Solutions to In-Class Problems Week 10, Wed. Problem 1. The Tao of BOOKKEEPER: we seek enlightenment through contemplation of the word BOOKKEEPER . (a) In how many ways can you arrange the letters in the word POKE ? Solution. There are 4! arrangements corresponding to the 4! permutations of the set { P, O, K, E } . (b) In how many ways can you arrange the letters in the word BO 1 O 2 K ? Observe that we have subscripted the O’s to make them distinct symbols. Solution. There are 4! arrangements corresponding to the 4! permutations of the set { B, O 1 , O 2 , K } . (c) Suppose we map arrangements of the letters in BO 1 O 2 K to arrangements of the letters in BOOK by erasing the subscripts. Indicate with arrows how the arrangements on the left are mapped to the arrangements on the right. O 2 BO 1 K KO 2 BO 1 BOOK O 1 BO 2 K OBOK KO 1 BO 2 KOBO BO 1 O 2 K BO 2 O 1 K . . . . . . (d) What kind of mapping is this, young grasshopper? Solution. 2-to-1 (e) In light of the Division Rule, how many arrangements are there of BOOK ? Solution. 4! / 2 (f) Very good, young master! How many arrangements are there of the letters in KE 1 E 2 PE 3 R ? Solution. 6! Creative Commons 2010, Prof. Albert R. Meyer .
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2 Solutions to In-Class Problems Week 10, Wed. (g) Suppose we map each arrangement of KE 1 E 2 PE 3 R to an arrangement of KEEPER by eras- ing subscripts. List all the different arrangements of KE 1 E 2 PE 3 R that are mapped to REPEEK in this way. Solution. RE 1 PE 2 E 3 K , RE 1 PE 3 E 2 K , RE 2 PE 1 E 3 K , RE 2 PE 3 E 1 K , RE 3 PE 1 E 2 K , RE 3 PE 2 E 1 K (h) What kind of mapping is this?
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