test1-solutions - UNIVERSITY OF VICTORIA MATHEMATICS 151,...

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Unformatted text preview: UNIVERSITY OF VICTORIA MATHEMATICS 151, SECTION A02, FALL 2008 MIDTERM 1 SOLUTIONS, OCTOBER 10 Question 1. A student must answer exactly 15 out of 20 questions on an exam. If the student must answer at least 8 of the first 10 questions, how many choices does the student have for which questions they answer? (A) 150 (B) 300 (C) 600 (D) 1 , 200 (E) 2 , 400 (F) 4 , 800 (G) 9 , 600 (H) 19 , 200 (I) 38 , 400 (J) 76 , 800 Solution: The student can either answer 8 , 9, or 10 of the first 10 questions. This can be done in 10 8 10 7 + 10 9 10 6 + 10 10 10 5 = (45)(120) + (10)(210) + (252) = 7 , 752 ways. So the answer is (G). Question 2. If there is a party where every person shakes hands with every other person exactly once, and there are 91 handshakes in total, how many people are there at the party? (Note: A handshake between any 2 people is considered a single handshake.) (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 (F) 12 (G) 13 (H) 14 (I) 15 (J ) 16 Solution: For each handshake, we choose two people to be involved. There are thus 91 ways to choose two people. We check to see that 91 is equal to ( 14 2 ) , and so there are 14 people at the party. The answer is therefore (H). Question 3. How many ways can the letters in CAPITALIZE be arranged so that all of the con- sonants are kept together? (A) 200 (B) 400 (C) 800 (D) 1 , 600 (E) 3 , 200 (F) 6 , 400 (G) 12 , 800 (H) 25 , 600 (I) 51 , 200 (J) 102 , 400 Solution: Putting the consonants in a box, we have 6 things to arrange, then 5 in the box to arrange, with 2 pairs of duplicated letters. This gives 6!5! 2!2! = 21 , 600. So the answer is (H). Question 4. Following a victory, a baseball team has a 65% chance of winning their next game....
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test1-solutions - UNIVERSITY OF VICTORIA MATHEMATICS 151,...

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