Problem Set 1 Solutions

# Problem Set 1 Solutions - Head-counting 40C 35E 30M 25P(a...

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CSE 21: Solutions - Problem Set 1 1. There are 3 patterns: CVCVCC, CVCCVC, CCVCVC, where C stands for consonant and V for vowel. Note that each pattern has two adjacent consonants, which can not be the same. There are 3 · 2 · 3 · 2 · (3 · 2) = 216 names for each pattern. Then there are 3 · 216 = 648 names in total. 2. B - boy, G - girl (a) The problem is equivalent to arranging 7 people in a row. There are 7! = 5040 ways to do this. (b) There are only two patterns: BBBBGGG, GGGBBBB. The number of ways is 2 · (3! · 4!) = 288 . (c) There is only one pattern: BGBGBGB. The number of ways is 4! · 3! = 144 . 3. head-counting: 4S, 3A, 1F, 1R. Four distinct letters. (a) 4! = 24. (b) 4 · 4 · 4 = 64 (c) There are 5 patterns: ♣♣♣ , ♣♣♦ , ♣♦♣ , ♦♣♣ , ♣♦♥ , where diﬀerent symbols stand for diﬀerent letters. The ﬁrst pattern has only two words: SSS, AAA. For each of the next three patterns, there are 2 · (4 - 1) = 6 words. The last pattern has 4 · 3 · 2 = 24 words. Thus the total number of words is 2 + 3 * 6 + 24 = 44 . 4. C - cse, E - ece, M -math, P -physics.
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Unformatted text preview: Head-counting: 40C, 35E, 30M, 25P. (a) There are 4 patterns: { C,E,M } , { C,E,P } , { C,M,P } , { E,M,P } . Then 40 Â· 35 Â· 30 + 40 Â· 35 Â· 25 + 40 Â· 30 Â· 25 + 35 Â· 30 Â· 25 = 133250 . (b) Consider the complement: if all three members are from the same department, there are Â± 40 3 Â² + Â± 35 3 Â² + Â± 30 3 Â² + Â± 25 3 Â² = 22785 choices. The number of choices without any restriction is Â± 40 + 35 + 30 + 25 3 Â² = 357760 . Thus the answer for this part is 357760-22785 = 334975 . (c) There are two patterns: { C,M,P } , { E,M,P } . The answer is 40 Â· 30 Â· 25 + 35 Â· 30 Â· 25 = 56250 . 5. Consider the complement : passwords that contain no digit or no letter, in other words, contain only letters or digits. There are 5 6 = 15625 passwords that contain only letters. Similarly, there are also 15625 passwords that contain only digits. The total number of passwords is 10 6 . Thus our answer is 10 6-2 Â· 15625 = 968750 . 1...
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