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MIT6_042JF10_rec15_sol

MIT6_042JF10_rec15_sol - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science November 3, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 15 1 The Tao of BOOKKEEPER In this problem, we seek enlightenment through contemplation of the word BOOKKEEPER . 1. 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 } . 2. 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 } . 3. 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 O 1 BO 2 K KO 1 BO 2 BO 1 O 2 K BOOK OBOK KOBO BO 2 O 1 K . . . . . . 4. What kind of mapping is this, young grasshopper? Solution. 2-to-1 5. In light of the Division Rule, how many arrangements are there of BOOK ? Solution. 4! / 2 6. Very good, young master! How many arrangements are there of the letters in KE 1 E 2 PE 3 R ?
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2 Recitation 15 Solution. 6! 7. Suppose we map each arrangement of KE 1 E 2 PE 3 R to an arrangement of KEEPER by erasing 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 8. What kind of mapping is this? Solution. 3!-to-1 9. So how many arrangements are there of the letters in KEEPER ? Solution. 6! / 3! 10. Now you are ready to face the BOOKKEEPER! How many arrangements of BO 1 O 2 K 1 K 2 E 1 E 2 PE 3 R are there? Solution. 10! 11. How many arrangements of BOOK 1 K 2 E 1 E 2 PE 3 R are there? Solution. 10! / 2! 12. How many arrangements of BOOKKE 1 E 2 PE 3 R are there? Solution. 10! / (2! 2!) · 13. How many arrangements of BOOKKEEPER are there? Solution. 10! / (2! 2! 3!) · · 14. How many arrangements of V OODOODOLL are there? Solution. 10! / (2! 2! 5!) · · 15. (IMPORTANT) How many n -bit sequences contain k zeros and ( n k ) ones? Solution. n ! / ( k ! ( n k )!) · This quantity is denoted n k and read n choose k ”. You will see it almost every day in 6.042 from now until the end of the term. Remember well what you have learned: subscripts on, subscripts off. This is the Tao of Bookkeeper.
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Recitation 15 3 2 Pigeonhole Principle Solve the following problems using the pigeonhole principle. For each problem, try to identify the pigeons , the pigeonholes , and a rule assigning each pigeon to a pigeonhole.
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