CountingExamples.pdf

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1. Example 1: In how many ways can we select two books from different subjects among five CS books, three Math and two Art books? Select CS+Math, CS+Art, Math+Art: 5 × 3 + 5 × 2 + 2 × 3 = 15+10+6 = 31 ways Example 3: There are n senators. In how many ways can subcommittees be formed that consist of at least 2 and at most n-1 senators? 2n – n – 1 -1 (don’t forget the empty set) 1
2. Combinations Example: A robot moves in a 6 by 6 grid from A to B. How many district ways are there for a robot to move from the upper left corner A to the lower right corner B when movement is right or down? 2 A B
3. Combinations Example: How many district ways are there for moving from A to B travelling right and down? Solution: Every path consists 6 right moves and 6 down moves. A path is a string of length 12: RRRRDRDDRDDD The total number of paths is the number of 6- combinations of a set with 12 elements 12,6 = 12! 6! 6! 3 A B
4. Combinations Example: A robot moves in a 4 by 10 grid from A to B. How many district ways are there for a robot to move from the upper left corner A to the lower right corner B when movement is right or down? 4 A B A. C(14,10) B. C(10,4) C.C(14,4) D.C(18,10) E. 14!
5. Example: Hands with a Full House A Full House is a hand with three cards of one rank and 2 cards of another rank. In a 52 card deck, how many Full house hands are there? Solution: 5
6. Example: Hands with a Full House A Full House is a hand with three cards of one rank and 2 cards of another rank. In a 52 card deck, how many Full house hands are there? Solution: ▪ Rank of triple can be chosen in 13 ways ▪ Suit of triple can be selected in 4 3 ways ▪ Rank of pair can be chosen in 12 ways ▪ Suit of pair can be selected in 4 2 ways Number of Full Houses is 13 × 4 3 × 12 × 4 2 = 3,744 6
7. Example: Martians, Vesuvians and Jovians In how many ways can five distinct Martians, 6 distinct Vesuvians, and four distinct Jovians wait in line if no two Martians stand together? M1 M2 M3 M4 M5 V1 V2 V3 V4 V5 V6 J1 J2 J3 J4 V1 V2 M1 V3 V4 M2 V5 J1 M3 J2 M4 J3 J4 V6 M6 7
8. Example: Martians, Vesuvians and Jovians In how many ways can five distinct Martians, six distinct Vesuvians, and four distinct Jovians wait in line if no two Martians stand together? V1 V2 M1 V3 V4 M2 V5 J1 M3 J2 M4 J3 J4 V6 M6 Solution: Arrange Vesuvians and Jovians in line. You can arrange them in 10! ways. As no two Martians can stand together, each Martian should stand in the 11 spaces in between Vesuvians and Jovians. The number of ways Martians can be arranged is P(11,5). Total number of ways is 10! × P(11,5) 8
9. Example: Generalized Pigeonhole Principle The principal’s office in Pigeonhole Elementary has 50 bags, one for each class. The bags contain confiscated cell phones. Each bag contains between 0 and 23 phones. Show that there are at least 3 bags containing the same number of phones. Solution: Who are the pigeons and who are the holes? 9
10. Example: Generalized Pigeonhole Principle The principle’s office in Pigeonhole Elementary has 50 bags, one for each class. The bags contain confiscated cell phones. Each bag contains at most 23 phones. Show that there are at least 3 bags containing the same number of phones. Solution: There are 24 pigeonholes and 50 pigeons (the bags). – Place the bags into the pigeonhole matching the number of phones the bag contains (a number between 0 and 23). – Then, there exists one pigeonhole that contains at least ⌈50/24⌉ = 3 bags. Implies that at least 3 bags contain the same number of phones. 10