quiz5 sol - such decimal digits c There are C(3 2 =3 ways...

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Name__________________________ UFID#_______________ COT 3100 – SPRING 2010 Quiz #5 For the following two questions, please justify your answers. Answers with only a number (without explanation) will receive zero point. Question 1: (6pts) How many strings of three decimal digits: a) Do not contain the same digit twice? b) End with an even digit? c) Have exactly two digits that are 5s? a) If we want to make the digits not contain the same digit twice. Then, the first digit has 10 ways to choose; the second digit has 9 ways to choose; the third digit has 8 ways to choose. So, there are ± ² ³ ² ´ µ ¶·±¸ decimal digits. b) We only need to make sure the last digit is: 0, 2, 4, 6 or 8. Then, totally, there are ± ² ± ² ¹ µ ¹±±
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Unformatted text preview: such decimal digits. c) There are C (3, 2) =3 ways to choose the position of the two 5s. There are 9 ways (we cannot choose 5) to choose the other number. So totally, there are º ² ³ µ ·¶¸ such decimals. Question 2: (4pts) How many numbers must be selected from the set {1, 3, 5, 7, 9, 11, 13, 15} to guarantee that at least one pair of these numbers add up to 16? Consider the 4 groups {1, 15}, {3, 13}, {5, 11} and {7, 9}. If there are 5 or numbers, then there must be two numbers from the same group. Thus, the sum of them is 16. If there are only 4 or less numbers, then there is a possibility that none of them are from the same group. Then the sum is not 16 for any pair. Therefore, 5 is sufficient and necessary....
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