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Math 15B (Bender) Midterm Exam 27 October 2000 1. (36 pts) A five digit number is a sequence of five digits, the first of which is NOT zero. Thus, 12345 and 10101 are valid but 01234 and 1234 are NOT valid. (a) Ans: (b) Ans: (c) Ans: How many five digit numbers are there? One way is to note there are 9 choices for the first digit and 10 for each of the remaining giving 9 x 10%. You could also note that it is simply all numbers less than 10° and at least 10? giving 10° 10* = 9 x 10". How many five digit numbers have all digits different? (as in 12345 but not 10101) There are 9 choices for the first digit, 9 for the second (anything but the first), 8 for the third (anything but the first two), and so on, giving 9 x 9 x 8 x 7 x 6. How many five digit numbers have no digit appearing just once? (So 11111 and 10010 are okay, but 10111 and 12312 are not.) Either one digit appears 5 times OR two digits appear, one twice and one three times. There are 9 possibilities for the first case. For the second case, choose the two positions for the pair of digits in (2') ways, choose the first digit in the number in 9 ways (anything but zero), and choose the other digit in 9 ways. Thus we have 9 + (3) x 9 x9. 2. (16 pts.) Nine people, including Alice, are to be divided into two teams of four people each, plus a referee. If all divisions are equally likely, what is the probability that Alice is the referee? (No, it doesn't matter if the teams are distinguishable or not.) Be sure to explain how you got your answer. Ans: There are nine choices for the referee, all of which are equally likely. Thus, the probability that Alice is the referee is 1/9. 3. (24 pts.) An integer k from 1 to 9 is picked uniformly at random. Let X(k) =1if k is odd and X (k) = 0 if k is even. Let Y (k) be the remainder when k is divided by 3. (a) Ans: Draw a table like the one here and fill in the probabilities. First we construct a table of X and Y for each possible outcome. Since each outcome has probability 1/9, a simple count lets us fill the table. 123456789 X\Yl o 1 2 X[101010101 0 1/9 1/9 2/9 Y [120120120 1 2/9 2/9 1/9
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