# Hmk4Sol - COT 3100H Spring 2008 Homework#4 Solutions 1 The...

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COT 3100H Spring 2008 Homework #4 Solutions 1) The House currently has 199 Republicans, 232 Democrats and four vacancies. (Nathan corrected my information.) Answer the following questions about forming committees in the House. (a) How many committees can be formed with 5 Congresspersons? 5 431 , since we are choosing 5 people out of 431. (b) How many committees can be formed with 3 Republicans and 2 Democrats? 2 232 3 199 , since we are choosing 3 people out of 199, and 2 out of 232. Since each combination of 3 can be paired up with each combination of 2, we are counting a cartesian product between the two sets, so we should multiply their cardinalities. (c) How many committees of 4 Congresspersons have Tom Feeney (Representative that Nathan works for)? We can only choose 3 people since the fourth has been chosen for us. Now, we are choosing out of 430 remaining senators, so the answer is . 3 430 (d) How many committees of 20 Congresspersons have at least 3 Democrats? This question is easier to do using the subtraction principle. We will first count the number of committees with 20 Congresspersons, and subtract out all the committees with 0, 1 or 2 Democrats. - - - 2 232 18 199 1 232 19 199 20 199 20 431 . The first two terms are of the form of answers from (a), since we are just choosing a subset of people from one group. The last two terms are the form of the answer to (b), because each of these terms includes a set number of republicans and democrats.

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2) The following questions will deal with permutations of the city name ADDISABABA. (a) How many total permutations are there?
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