Lesson Upper level problems on a party of people sitting at a round table
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<H2>Upper level problems on a party of people sitting at a round table</H2> <H3>Problem 1</H3>At a dinner party there are 10 people. They all sit at a round table. In how many ways can they sit if neither Amy nor Bob want to sit next to Carl? <B>Solution</B> <pre> In such problems, the indistinguishable arrangements are those that are obtained one from the other by a circular rotation. For 10 people around a round table, there are 9! distinguishable arrangements (= a universal set U of arrangements). Of them, there are 8! arrangements, where A sits next to C on the left of him and 8! arrangements, where A sits next to C on the right of him. So, in the universal set U, there are 2*8! arrangements {A}, where A sits next to C. Similarly, in the universal set U, there are 2*8! arrangements {B}, where B sits next to C. These arrangements, {A} and {B}, have non-empty intersection. This intersection has 7! arrangements {ACB} and 7! other arrangements {BCA}. From it, we conclude that the number of favorable arrangements is 9! - 2*8! - 2*8! + 2*7! = 1*2*3*4*5*6*7*8*9 - 4*(1*2*3*4*5*6*7*8) + 2*(1*2*3*4*5*6*7) = 211680. <U>ANSWER</U> </pre> <H3>Problem 2</H3>At a meeting, 4 scientists, 3 mathematicians, and 2 journalists are to be seated around a circular table. How many different circular arrangements are possible if every mathematician must sit next to a journalist? <pre> Draw a circle - it will represent the circular table. There are 9 chairs around the table. Let assume that the chairs are numbered from 1 to 9 sequentially clockwise and let assume that the chair #1 is in position North, or 12 o'clock. We will place one of the two journalist at the chair #1. Then the other journalist can not occupy neither of two neighbor chairs, since otherwise there is no place for 3 mathematicians next to two journalists. It means that the other journalist can occupy any one chair from #3 to #8 inclusive. Thus, there are 9 - 3 = 6 possibilities for the other journalist's chair. OK. So, there are 2*6 = 12 possibilities to place two journalists. (1) Next, assume that two journalists are just placed this way. Then there are 4 places neighbor these two journalists chairs to place 3 mathematician there. There are {{{C[4]^3}}} = 4 ways to place 3 mathematicians at these 4 chairs. (2) So, now we have 2+3 = 5 chairs occupied and 9-5 = 4 chairs free for four scientists. They can be placed in 4! = 24 different ways in these 4 chairs. (3) Now calculate the product of options n = 12 (from (1)) * 4 (from (2)) * 24 (from (3)) = 12 * 4 * 24 = 1152. At this point, the problem is solved to the end, and the number of all different arrangements is 1152. <U>ANSWER</U> </pre> My other additional lessons on Combinatorics problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-league-combinatorics-problem.lesson>Upper league combinatorics problem</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-level-combinatorics-problem-on-subsets-of-a-finite-set.lesson>Upper level combinatorics problem on subsets of a finite set</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/A-confusing-combinatorics-problem-on-repeating-digits.lesson>A confusing combinatorics problem on repeating digits in numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-level-combinatorics-problems-on-Inclusion-Exclusion-principle.lesson>Upper level combinatorics problems on Inclusion-Exclusion principle</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/This-nice-problem-teaches-to-distinguish-permutations-from-combinations.lesson>This nice problem teaches to distinguish permutations from combinations</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-of-additional-combinatorics-problems.lesson>OVERVIEW of additional combinatorics problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
