MIT6_042JF10_rec15_sol

MIT6_042JF10_rec15_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 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 Os 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 ? 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....
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.

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MIT6_042JF10_rec15_sol - 6.042/18.062J Mathematics for...

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