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15b_00f_ms

# 15b_00f_ms - Math 15B(Bender Midterm Exam 27 October 2000...

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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) How many ﬁve digit numbers are there? Ans: One way is to note there are 9 choices for the ﬁrst digit and 10 for each of the remaining giving 9 × 10 4 . You could also note that it is simply all numbers less than 10 5 and at least 10 4 giving 10 5 - 10 4 =9 × 10 4 . (b) How many ﬁve digit numbers have all digits diﬀerent? (as in 12345 but not 10101) Ans: 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 × 9 × 8 × 7 × 6. (c) How many ﬁve digit numbers have no digit appearing just once? (So 11111 and 10010 are okay, but 10111 and 12312 are not.) Ans: 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 ( 5 2 )

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15b_00f_ms - Math 15B(Bender Midterm Exam 27 October 2000...

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