PS01-solutions - MATH 681 Problem Set #1 solutions 1. (10...

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Unformatted text preview: MATH 681 Problem Set #1 solutions 1. (10 points) A five-digit number must not have a zero as its first digit. (a) (5 points) How many five-digit numbers are there whose digits appear in increas- ing order (e.g., 25689)? This is a matter of choosing five distinct nonzero digits without regard to or- der: once the five digits are chosen, the order is necessarily determined, i.e. { 2 , 5 , 6 , 8 , 9 } can only be ordered as 25689; zero is forbidden since, as the smallest single digit, it would by necessity be first, which would not result in a five-digit number. Thus, the five-digit numbers with digits in increasing order can be placed into a one-to-one correspondence with the 5-element subsets of { 1 , 2 , 3 ,..., 9 } . There are ( 9 5 ) = 126 such sets. (b) (5 points) How many five-digit numbers are there whose digits appear in descend- ing order (e.g., 98652)? This situation is exactly as above, except that zero is not forbidden, since as the smallest digit it will appear in the last position. Thus there are ( 10 5 ) = 252 such numbers. 2. (10 points) The members of two committees are being drawn from a pool of eight people. Each person can be on either committee or on no committee at all....
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PS01-solutions - MATH 681 Problem Set #1 solutions 1. (10...

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