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The university of illinois math 413 test 1 from spring 2010. The test includes various mathematical problems and their solutions. The problems cover topics such as combinatorics, number theory, and probability. Students are required to show their work and write their names on the test. No external aids are allowed during the test.
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University of Illinois MATH 413 Test 1 – Spring 2010
Answer as many problems as you can. Each question is worth 6 points (total points is 30). Show your work. An answer with no explanation will receive no credit. Write your name on the top right corner of each page. No external aids (notes, calculators, ipods etc) allowed. [Total time: 50 minutes]
1
Solution: Let x 1 , x 2 ,... , x 6 represent the number of each of the pastries Lewis buys. The problem asks to solve
x 1 + x 2 + · · · + x 6 = 12
subject to x 1 ≥ 1, x 2 ≥ 1, x 3 ≥ 1, x 4 ≥ 0, x 5 ≥ 0 and x 6 ≥ 0. Let yi = xi − 1 for i = 1, 2 , 3 and yi = xi for i = 4, 5 , 6. Then it is equivalent to count solutions to
y 1 + y 2 + · · · + y 6 = 9
subject to yi ≥ 0 for all i. The number of solutions of this is
6 − 1
14 5
(^1) Any similarity in names used in this test to people in the class is purely
coincidental.
Solution: There are 6 × 9 = 54 possible ways to pick letters. One can only obtain the same letter if we choose “A” or “N”. If it’s an A there are 4 ways to do it, and if it’s an N there is only 1. Hence the probability is 5/54.
Solution: Let ai be the number of hours he spends total on Facebook on days 1, 2 ,... , i. Note that
1 ≤ a 1 < a 2 <... < a 63 ≤ 90.
The fact 1 ≤ a 1 is because he spent at least one hour on day one. The claim ai < ai+1 holds since he spends at least one each day. Finally, there are 63 = 9 × 7 days in nine weeks. The fact a 63 ≤ 90 holds is since he spends at most 10 hours per week for the nine weeks. Consider 11 ≤ a 1 + 35 < a 2 + 35 <... < a 63 + 35 ≤ 125.
By the pigeonhole principle, at least two of the 126 numbers {ai} and {ai + 35} are the same. Clearly none of the ai’s are the same and none of the (ai + 35)’s are the same. Hence aj = ai +35 for some j ≥ i. Then on days i, i+1,... , j he was on Facebook exactly 35 hours.
(^2) He has to also spend time on his Myspace and Twitter accounts, and also
comment “First!” as many times as possible at TMZ.com, but never mind that, LOL
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