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Counting examples

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Counting examples

1. 1. Counting Examples Mathematics 4 February 14, 2012Mathematics 4 () Counting Examples February 14, 2012 1 / 13
2. 2. Fundamental Counting PrincipleIf one event can be done independently n diﬀerent ways and a secondevent can be done independently m diﬀerent ways, the total number ofways all things may be done in the stated order is n · m. Mathematics 4 () Counting Examples February 14, 2012 2 / 13
3. 3. Example 1 How many three-digit numbers can be formed from the digits 1, 2, 3, 4, and 5 if repetitions are allowed? If repetitions are not allowed? Mathematics 4 () Counting Examples February 14, 2012 3 / 13
4. 4. Example 2 How many numbers can be formed from the digits 1, 2, 3, 4, and 5 if repetitions are not allowed? (1 to 54321) Mathematics 4 () Counting Examples February 14, 2012 4 / 13
5. 5. Example 3If eight basketball teams are in a tournament, ﬁnd the number of diﬀerent ways that ﬁrst, second, and third place can be decided, assuming ties are not allowed. Mathematics 4 () Counting Examples February 14, 2012 5 / 13
6. 6. Example 4A girl has four skirts, six blouses, and three sweaters. How many diﬀerent skirt-blouse-sweater combinations can she wear? Mathematics 4 () Counting Examples February 14, 2012 6 / 13
7. 7. Example 5Automobile license plates start with three letters of the alphabet, followed by three digits. Find how many diﬀerent license plates are possible if 1 the ﬁrst digit following the last letter cannot be 0 Mathematics 4 () Counting Examples February 14, 2012 7 / 13
8. 8. Example 5Automobile license plates start with three letters of the alphabet, followed by three digits. Find how many diﬀerent license plates are possible if 1 the ﬁrst digit following the last letter cannot be 0 2 the ﬁrst letter cannot be I or O and the ﬁrst digit cannot be 0 Mathematics 4 () Counting Examples February 14, 2012 7 / 13
9. 9. Example 6Two diced are tossed, one after the other. In how many diﬀerent ways can they fall? Mathematics 4 () Counting Examples February 14, 2012 8 / 13
10. 10. Example 7 Two diced are tossed, one after the other. How many diﬀerent ways can the sum of the dots equal 1 3 Mathematics 4 () Counting Examples February 14, 2012 9 / 13
11. 11. Example 7 Two diced are tossed, one after the other. How many diﬀerent ways can the sum of the dots equal 1 3 2 5 Mathematics 4 () Counting Examples February 14, 2012 9 / 13
12. 12. Example 7 Two diced are tossed, one after the other. How many diﬀerent ways can the sum of the dots equal 1 3 2 5 3 7 Mathematics 4 () Counting Examples February 14, 2012 9 / 13
13. 13. Example 7 Two diced are tossed, one after the other. How many diﬀerent ways can the sum of the dots equal 1 3 2 5 3 7 4 9 Mathematics 4 () Counting Examples February 14, 2012 9 / 13
14. 14. Example 7 Two diced are tossed, one after the other. How many diﬀerent ways can the sum of the dots equal 1 3 2 5 3 7 4 9 5 11 Mathematics 4 () Counting Examples February 14, 2012 9 / 13
15. 15. Example 8 A row of six seats in a classroom is to be ﬁlled by selecting individuals from a group of six boys and four girls. If boys and girls are to be alternated, ﬁnd the number of diﬀerent seating arrangements. Mathematics 4 () Counting Examples February 14, 2012 10 / 13
16. 16. Example 9In how many ways can a test consisting of ten true-or-false questions can be completed? Mathematics 4 () Counting Examples February 14, 2012 11 / 13
17. 17. Example 10 A test consists of six multiple-choice questions, and there are ﬁve choicesfor each question. In how many diﬀerent ways can the test be completed? Mathematics 4 () Counting Examples February 14, 2012 12 / 13
18. 18. Example 11 Three high-school couples have purchased movie tickets. Partners are tobe seated next to each other, and the six seats are in a row. In how many ways can the six people be seated? Mathematics 4 () Counting Examples February 14, 2012 13 / 13
19. 19. Example 12 In how many ways can a standard deck of cards be shuﬄed so that the four aces appear on the top of the deck? Mathematics 4 () Counting Examples February 14, 2012 14 / 13
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