mt1 - MATH 3012 Applied Combinatorics(Fall07 TEST...

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MATH 3012 Applied Combinatorics (Fall’07) – TEST 1 (Solutions) Instructor : Prasad Tetali, office: Skiles 234, email: [email protected] Time: 1 hour 25 minutes Total Score: 5 + 10 + 10 + 5 + 10 = 40 pts 1 (5pts). A license plate has 3 letters (from the English alphabet) followed by 3 digits from 0 through 9. How many license plates can be made? (Assume that letters can not be repeated, but digits can be repeated.) Solution . Note that ordering matters, and that letters can not be reused. So it is 26 × 25 × 24 × 10 × 10 × 10. 2 (10pts). If n Z + and n 2, prove that 2 n < 2 n n < 4 n . Solution . It is simplest to prove by induction on n 2. Base Case : True, since 2 2 < 4 2 = 6 < 4 2 . Induction Hypothesis . Assume that for some n = k , k 2, the inequalities are true. Induction Step . Need to prove the case of n = k + 1, namely that 2 k +1 < 2( k +1) k +1 < 4 k +1 . Observe that 2( k + 1) k + 1 = (2 k + 2)(2 k + 1)(2 k )! ( k + 1)( k + 1) k ! k ! = 2 (2 k + 1) ( k + 1) 2 k k ( * ) Hence we may say, 2 k +1 = 2 2 k < 2 2 k k by the induction hypothesis 2 (2 k + 1) k + 1 2 k k since 1 (2 k +1) ( k +1) = (2 k + 1) k + 1
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