Discrete Mathematics with Graph Theory (3rd Edition) 187

Discrete Mathematics with Graph Theory (3rd Edition) 187 -...

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Section 7.2 185 16. (a) [BB] 9 x 9 x 8 x 7 x 6 x 5 x 4 = 9 x P(9,6) = 9 x 60,480 = 544,320. (b) The total number of 7 -digit numbers which do not contain a 6 is 8 x 8 x 7 x 6 x 5 x 4 x 3 = 161,280. The number which contain neither a 3 nor a 6 is 7 x 7 x 6 x 5 x 4 x 3 x 2 = 35,280. Hence, the answer is 161,280 - 35,280 = 126,000. 17. (a) P(9,7) = 9 x 8 x 7 x 6 x 5 x 4 x 3 = 181,440. (b) [BB] There are seven available positions for the 3, then six for the 6, then five for the remaining seven digits. The answer is 7 x 6 x P(7, 5) = 42 x 2520 = 105,840. (c) There are six pairs of consecutive positions into each of which we can place 36 or 63. After this, there remain five positions to be filled from the remaining seven digits. The answer is 6 x 2 x P(7,5) = 30,240. (d) [BB] There are seven digits to be placed in seven positions. The answer is P(7, 7) = 5040. (e) There are seven positions for the 3, then seven numbers to be placed in six positions. The answer is 7 x P(7,6) = 35,280. (t) Using the result of the previous part, 35280
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