Lesson OVERVIEW of additional combinatorics problems

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<H2>OVERVIEW of additional combinatorics problems</H2>

My additional lessons on Combinatorics problems in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-league-combinatorics-problem.lesson>Upper league combinatorics problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Permutations/An-upper-level-problem-on-a-party-of-people-sitting-at-a-round-table.lesson>Upper level problems on a party of people sitting at a round table</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-level-combinatorics-problem-on-finding-the-number-of-arrangements-along-a-straight-line.lesson>Upper level combinatorics problem on finding the number of arrangements along a straight line</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

<H3>List of my additional lessons on Combinatorics problems with short annotations</H3>

<A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-league-combinatorics-problem.lesson>Upper league combinatorics problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;From a sample of &nbsp;800 &nbsp;consumers, &nbsp;230 &nbsp;took coffee, &nbsp;245 &nbsp;took tea and &nbsp;325 &nbsp;took cocoa.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;30 &nbsp;took all the three beverages, &nbsp;70 &nbsp;took coffee and cocoa, &nbsp;110 &nbsp;took coffee only and &nbsp;185 &nbsp;took cocoa only.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(a) &nbsp;&nbsp;Find the number of consumers who took coffee and tea only.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(b) &nbsp;&nbsp;Find the number of consumers who took tea and cocoa only.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(c) &nbsp;&nbsp;Find the number of consumers who took tea only.

<A HREF=https://www.algebra.com/algebra/homework/Permutations/An-upper-level-problem-on-a-party-of-people-sitting-at-a-round-table.lesson>Upper level problems on a party of people sitting at a round table</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;At a dinner party there are &nbsp;10 &nbsp;people. &nbsp;They all sit at a round table.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In how many ways can they sit if neither &nbsp;Amy nor &nbsp;Bob want to sit next to &nbsp;Carl?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;At a meeting, &nbsp;4 &nbsp;scientists, &nbsp;3 &nbsp;mathematicians, &nbsp;and &nbsp;2 &nbsp;journalists are to be seated around a circular table.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many different circular arrangements are possible if every mathematician must sit next to a journalist?

<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>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;S = &nbsp;{1, 2, 3, 4, . . . , n}, &nbsp;and let &nbsp;A &nbsp;and &nbsp;B &nbsp;be two subsets of &nbsp;S &nbsp;such that &nbsp;A ≠ ∅, &nbsp;B ≠ S, &nbsp;and &nbsp;A ⊆ B.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculate the number of ordered pairs &nbsp;(A,B) &nbsp;for all subsets of the set &nbsp;S.

<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>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;What is the number of different five-digit numbers that can be formed

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;from the set &nbsp;&nbsp;S = {2,3,4,5,6} &nbsp;&nbsp;such that one digit is repeated twice and another digit is repeated twice.

<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>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;In how many ways can we seat &nbsp;3 &nbsp;pairs of siblings in a row of &nbsp;10 &nbsp;chairs,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;so that in each pair the siblings seat together, &nbsp;but different pairs are not next to each other ?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In how many ways can we seat &nbsp;3 &nbsp;pairs of siblings in a row of &nbsp;10 &nbsp;chairs,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;so that nobody sits next to their sibling ?  &nbsp;&nbsp;(Four chairs will be left empty, &nbsp;of course.)

<A HREF=https://www.algebra.com/algebra/homework/Permutations/Upper-level-combinatorics-problem-on-finding-the-number-of-arrangements-along-a-straight-line.lesson>Upper level combinatorics problem on finding the number of arrangements along a straight line</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the number of ways to arrange &nbsp;4 &nbsp;green balls, &nbsp;3 &nbsp;red balls, &nbsp;and &nbsp;2 &nbsp;white balls

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in a straight line such that no two balls of the same color are adjacent to each other.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the number of ways of arranging one &nbsp;A, &nbsp;two &nbsp;Bs, &nbsp;three &nbsp;Cs, &nbsp;and four &nbsp;Ds, &nbsp;so that

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;no two &nbsp;Bs &nbsp;are next to each other, &nbsp;no two &nbsp;Cs &nbsp;are next to each other, &nbsp;and no two &nbsp;Ds &nbsp;are next to each other.

<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>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A scholarship committee has &nbsp;$3000 &nbsp;to award this year and has &nbsp;12 &nbsp;qualified candidates.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Thom thinks that individual awards of &nbsp;$1500, &nbsp;$1000, &nbsp;and &nbsp;$500 &nbsp;would be appropriate,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;while &nbsp;Peter &nbsp;thinks that &nbsp;3 &nbsp;awards of &nbsp;$1000 &nbsp;each would seem logical.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Count the number of possible outcomes for each plan.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.