Discrete Mathematics with Graph Theory (3rd Edition) 175

# Discrete Mathematics with Graph Theory (3rd Edition) 175 -...

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Section 6.2 5. (a) [BB] 9 x 26 x 26 x 26 x 10 x 10 x 10 = 9 . 26 3 . 10 3 = 158,184,000. (b) [BB] 9 x 26 X 10 5 = 23,400,000. (c) [BB] 158,184,000 + 23,400,000 = 181,584,000. 173 6. (a) The first three letters can be chosen arbitrarily, but then last two are determined. So the number of palindromes is 26 3 = 17,576. (b) 26 4 = 456,976. (c) 26 + 26 + 26 2 + 26 2 + 26 3 + 26 3 + 26 4 + 26 4 = 950,508 (d) 26 10 7. (a) [BB] 13 x 6 x 2 x 4 = 624; 8. (a) 26; (b) 8; (c) 36; (b)[BB] 13 + 4 = 17; (d) 40. (c) [BB] 25. 9. [BB] (a) 4 x 4 = 16; (b) 4 x 4 x 4 = 64; (c) 16 + 64 = 80. 10. 8 x 8 x 9 x 10 x 10 x 10 x 10 = 5,760,000 11. [BB] 60 x 60 x 60 = 216,000 12. Choose a row and then one of the seven adjacent pairs of squares in that row, or choose a column and then one of the seven adjacent pairs of squares in that column. There are 8 x 7 + 8 x 7 = 112 adjacent pairs of squares. 13. [BB] 500 + (500)(499) + (500)(499)(498) = 1.24501 x 10 8 14. (a) 3 x 5 = 15; (b) 15 x 15 = 225; (c) 3 x 5 x 4 x 2 = 120. 15. The numbers required are of the form 25x, 52x, 2x5, 5x2, x25 or x52. There are seven numbers of each type since there are seven digits 1, 2, 4, 5, 6, 8, 9 available. By the addition rule, there are 7(6) = 42 numbers altogether. Not all of these are different, however. When x = 2 or x = 5, six numbers are counted twice. So the answer is 42 - 6 = 36 numbers.
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