Mathematics for Computer Science
14.2. Counting Sequences 553 14.2.1 The Product Rule TheProduct Rulegives the size of a product of sets. Recall that ifP 1 ;P 2 ...
Chapter 14 Cardinality Rules554 14.2.3 The Sum Rule Bart allocates his little sister Lisa a quota of 20 crabby days, 40 irritabl ...
14.3. The Generalized Product Rule 555 Thus, the length-six passwords are in the setFS^5 , the length-seven passwords are inF ...
Chapter 14 Cardinality Rules556 Prizes fortruly exceptionalCoursework Given everyone’s hard work on this material, the instructo ...
14.3. The Generalized Product Rule 557 14.3.1 Defective Dollar Bills A dollar bill isdefectiveif some digit appears more than on ...
Chapter 14 Cardinality Rules558 (^8) 0Z0Z0Z0Z (^7) Z0Z0m0Z0 (^6) 0Z0Z0Z0Z (^5) Z0Z0Z0Z0 (^4) 0a0Z0Z0Z (^3) Z0Z0Z0Z0 (^2) 0Z0Z0o0 ...
14.4. The Division Rule 559 3ŠD 6 permutations of the 3-element setfa;b;cg, which is the number we found above. Permutations wil ...
Chapter 14 Cardinality Rules560 (^8) 0Z0Z0Z0s (^7) Z0Z0Z0Z0 (^6) 0Z0Z0Z0Z (^5) Z0Z0Z0Z0 (^4) 0Z0Z0Z0Z (^3) Z0Z0Z0Z0 (^2) 0Z0Z0Z0 ...
14.4. The Division Rule 561 seatings. An equivalent way to say this is that two seatings yield the same arrange- ment when they ...
Chapter 14 Cardinality Rules562 14.5 Counting Subsets How manyk-element subsets of ann-element set are there? This question aris ...
14.5. Counting Subsets 563 But we know there arenŠpermutations of ann-element set, so by the Division Rule, we conclude that nŠD ...
Chapter 14 Cardinality Rules564 14.6 Sequences with Repetitions 14.6.1 Sequences of Subsets Choosing ak-element subset of ann-el ...
14.6. Sequences with Repetitions 565 Notice that there are 1 B, 2 O’s, 2 K’s, 3 E’s, 1 P, and 1 R in BOOKKEEPER. This leads to a ...
Chapter 14 Cardinality Rules566 that you play through: “You know? The Bookkeeper Rule? Don’t you guys know anything?” The Bookke ...
14.7. Counting Practice: Poker Hands 567 The Binomial Theorem explains why thenchooseknumber is called abinomial coefficient. Th ...
Chapter 14 Cardinality Rules568 Five-Card Draw is the number of 5-element subsets of a 52-element set, which is 52 5 ! D2;598;96 ...
14.7. Counting Practice: Poker Hands 569 14.7.2 Hands with a Full House AFull Houseis a hand with three cards of one rank and tw ...
Chapter 14 Cardinality Rules570 The rank of the second pair, which can be chosen in 12 ways. The suits of the second pair, whic ...
14.7. Counting Practice: Poker Hands 571 Whenever you use a mappingf WA!Bto translate one counting problem to another, check th ...
Chapter 14 Cardinality Rules572 The suit of the extra card, which can be selected in 4 ways. The rank of the extra card, which ...
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