Lesson OVERVIEW of additional combinatorics problems

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OVERVIEW of additional combinatorics problems

My additional lessons on Combinatorics problems in this site are

    - Upper league combinatorics problem
    - Upper level problems on a party of people sitting at a round table
    - Upper level combinatorics problem on subsets of a finite set
    - A confusing combinatorics problem on repeating digits in numbers
    - Upper level combinatorics problems on Inclusion-Exclusion principle
    - Upper level combinatorics problem on finding the number of arrangements along a straight line
    - This nice problem teaches to distinguish permutations from combinations

List of my additional lessons on Combinatorics problems with short annotations

Upper league combinatorics problem

    Problem 1. From a sample of 800 consumers, 230 took coffee, 245 took tea and 325 took cocoa.
                       30 took all the three beverages, 70 took coffee and cocoa, 110 took coffee only and 185 took cocoa only.
                           (a)   Find the number of consumers who took coffee and tea only.
                           (b)   Find the number of consumers who took tea and cocoa only.
                           (c)   Find the number of consumers who took tea only.

Upper level problems on a party of people sitting at a round table

    Problem 1. 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?

    Problem 2. 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?

Upper level combinatorics problem on subsets of a finite set

    Problem 1. Let S = {1, 2, 3, 4, . . . , n}, and let A and B be two subsets of S such that A ≠ ∅, B ≠ S, and A ⊆ B.
                       Calculate the number of ordered pairs (A,B) for all subsets of the set S.

A confusing combinatorics problem on repeating digits in numbers

    Problem 1. What is the number of different five-digit numbers that can be formed
                       from the set   S = {2,3,4,5,6}   such that one digit is repeated twice and another digit is repeated twice.

Upper level combinatorics problems on Inclusion-Exclusion principle

    Problem 1. In how many ways can we seat 3 pairs of siblings in a row of 10 chairs,
                       so that in each pair the siblings seat together, but different pairs are not next to each other ?

    Problem 2. In how many ways can we seat 3 pairs of siblings in a row of 10 chairs,
                       so that nobody sits next to their sibling ?   (Four chairs will be left empty, of course.)

Upper level combinatorics problem on finding the number of arrangements along a straight line

    Problem 1. Find the number of ways to arrange 4 green balls, 3 red balls, and 2 white balls
                       in a straight line such that no two balls of the same color are adjacent to each other.

    Problem 2. Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that
                       no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

This nice problem teaches to distinguish permutations from combinations

    Problem 1. A scholarship committee has $3000 to award this year and has 12 qualified candidates.
                       Thom thinks that individual awards of $1500, $1000, and $500 would be appropriate,
                       while Peter thinks that 3 awards of $1000 each would seem logical.
                       Count the number of possible outcomes for each plan.

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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