cs2022d13quiz06sol.pdf - Ma2201/CS2022 Quiz 0110 Discrete...

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Ma2201/CS2022 Discrete Mathematics D Term, 2013 Quiz 0110 Solutions DOUBLE DOWN QUIZ: Normal Double Down TRIPLE DOWN 1. ( 4 pts ) How many 144 letter strings of lower case letters L = { a, b, c, . . . , z } have either only vowels ( V = { a, e, i, o, u } ) in the the odd positions, start with 100 identical letters, or read the same forwards as backwards. Let A be the strings with vowels in the odd positions, B the strings which start with 100 identical letters, and C be the strings which read the same forwards and backwards. We want to compute A B C and use inclusion exclusion. | A | = 5 72 · 26 72 , since there are 72 positions with vowels and 72 having any letter. | B | = 26 45 , since the choice of the first letter forces the next 99, leaving 44 to be independently chosen. | C | = 26 72 , only the first 72 can be independently chosen. | A B | = 5 23 26 22 , since the first 100 letters must all be identical vowels, and the last 44 letters are half vowel, half normal. | A C | = 5 72 , if it read the same forwards and backwards, there must be vowels in all positions, and the first 72 determine the rest.
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