How to apply the principle of inclusion and exclusion in this problem

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
4
down vote

favorite












8 new employees are assigned to 3 different departments. How many differents ways of assigning the employees exist if each department has to receive at least one employee?



At first I thought to work with the departments as sets but I think that is not possible beacuse the problem does not say that the employees can work in one or more departmets. Is working with the departments as sets the right option?




combinatorics inclusion-exclusion




share|cite|improve this question



















  • see this wanswer also: math.stackexchange.com/questions/550256/…
    – Hector Blandin
    Jul 31 at 4:00














up vote
4
down vote

favorite












8 new employees are assigned to 3 different departments. How many differents ways of assigning the employees exist if each department has to receive at least one employee?



At first I thought to work with the departments as sets but I think that is not possible beacuse the problem does not say that the employees can work in one or more departmets. Is working with the departments as sets the right option?




combinatorics inclusion-exclusion




share|cite|improve this question



















  • see this wanswer also: math.stackexchange.com/questions/550256/…
    – Hector Blandin
    Jul 31 at 4:00












up vote
4
down vote

favorite









up vote
4
down vote

favorite











8 new employees are assigned to 3 different departments. How many differents ways of assigning the employees exist if each department has to receive at least one employee?



At first I thought to work with the departments as sets but I think that is not possible beacuse the problem does not say that the employees can work in one or more departmets. Is working with the departments as sets the right option?




combinatorics inclusion-exclusion




share|cite|improve this question











8 new employees are assigned to 3 different departments. How many differents ways of assigning the employees exist if each department has to receive at least one employee?



At first I thought to work with the departments as sets but I think that is not possible beacuse the problem does not say that the employees can work in one or more departmets. Is working with the departments as sets the right option?





combinatorics inclusion-exclusion





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 31 at 2:56









Jorge Chang

254




254











  • see this wanswer also: math.stackexchange.com/questions/550256/…
    – Hector Blandin
    Jul 31 at 4:00
















  • see this wanswer also: math.stackexchange.com/questions/550256/…
    – Hector Blandin
    Jul 31 at 4:00















see this wanswer also: math.stackexchange.com/questions/550256/…
– Hector Blandin
Jul 31 at 4:00




see this wanswer also: math.stackexchange.com/questions/550256/…
– Hector Blandin
Jul 31 at 4:00










2 Answers
2






active

oldest

votes

















up vote
4
down vote



accepted










The number of ways to map $8$ people into $3$ departments is $N(0)=3^8$. The number of ways to miss a particular department would be $binom322^8$. The number of ways to miss two particular departments would be $binom311^8$. The number of ways to miss three particular departments would be $binom300^8$.



Inclusion-Exclusion says there are
$$
3^8-binom322^8+binom311^8-binom300^8=5796
$$
ways to distribute $8$ people among $3$ departments with no empty departments.






share|cite|improve this answer




























    up vote
    1
    down vote













    This is equivalent to count the number of surjective maps from the set of employees $e_1,dots,e_8$ to the set of departments $d_1,d_2,d_3$. So you get
    $$S(8,3)times 3!=5796quad textways$$
    where $S(8,3)$ is the Stirling number of second kind.
    Recall that the condition each department has to receive at least one employee means that each map has to be surjective.




    The number of surjective maps from an $n$ elements set onto a $k$ elements set is given by:
    $$S(n,k)cdot k!$$







    share|cite|improve this answer























      Your Answer




      StackExchange.ifUsing("editor", function ()
      return StackExchange.using("mathjaxEditing", function ()
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      );
      );
      , "mathjax-editing");

      StackExchange.ready(function()
      var channelOptions =
      tags: "".split(" "),
      id: "69"
      ;
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function()
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled)
      StackExchange.using("snippets", function()
      createEditor();
      );

      else
      createEditor();

      );

      function createEditor()
      StackExchange.prepareEditor(
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: false,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      );



      );








       

      draft saved


      draft discarded


















      StackExchange.ready(
      function ()
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2867603%2fhow-to-apply-the-principle-of-inclusion-and-exclusion-in-this-problem%23new-answer', 'question_page');

      );

      Post as a guest

















      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote



      accepted










      The number of ways to map $8$ people into $3$ departments is $N(0)=3^8$. The number of ways to miss a particular department would be $binom322^8$. The number of ways to miss two particular departments would be $binom311^8$. The number of ways to miss three particular departments would be $binom300^8$.



      Inclusion-Exclusion says there are
      $$
      3^8-binom322^8+binom311^8-binom300^8=5796
      $$
      ways to distribute $8$ people among $3$ departments with no empty departments.






      share|cite|improve this answer

























        up vote
        4
        down vote



        accepted










        The number of ways to map $8$ people into $3$ departments is $N(0)=3^8$. The number of ways to miss a particular department would be $binom322^8$. The number of ways to miss two particular departments would be $binom311^8$. The number of ways to miss three particular departments would be $binom300^8$.



        Inclusion-Exclusion says there are
        $$
        3^8-binom322^8+binom311^8-binom300^8=5796
        $$
        ways to distribute $8$ people among $3$ departments with no empty departments.






        share|cite|improve this answer























          up vote
          4
          down vote



          accepted







          up vote
          4
          down vote



          accepted






          The number of ways to map $8$ people into $3$ departments is $N(0)=3^8$. The number of ways to miss a particular department would be $binom322^8$. The number of ways to miss two particular departments would be $binom311^8$. The number of ways to miss three particular departments would be $binom300^8$.



          Inclusion-Exclusion says there are
          $$
          3^8-binom322^8+binom311^8-binom300^8=5796
          $$
          ways to distribute $8$ people among $3$ departments with no empty departments.






          share|cite|improve this answer













          The number of ways to map $8$ people into $3$ departments is $N(0)=3^8$. The number of ways to miss a particular department would be $binom322^8$. The number of ways to miss two particular departments would be $binom311^8$. The number of ways to miss three particular departments would be $binom300^8$.



          Inclusion-Exclusion says there are
          $$
          3^8-binom322^8+binom311^8-binom300^8=5796
          $$
          ways to distribute $8$ people among $3$ departments with no empty departments.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 31 at 3:54









          robjohn♦

          258k25296612




          258k25296612




















              up vote
              1
              down vote













              This is equivalent to count the number of surjective maps from the set of employees $e_1,dots,e_8$ to the set of departments $d_1,d_2,d_3$. So you get
              $$S(8,3)times 3!=5796quad textways$$
              where $S(8,3)$ is the Stirling number of second kind.
              Recall that the condition each department has to receive at least one employee means that each map has to be surjective.




              The number of surjective maps from an $n$ elements set onto a $k$ elements set is given by:
              $$S(n,k)cdot k!$$







              share|cite|improve this answer



























                up vote
                1
                down vote













                This is equivalent to count the number of surjective maps from the set of employees $e_1,dots,e_8$ to the set of departments $d_1,d_2,d_3$. So you get
                $$S(8,3)times 3!=5796quad textways$$
                where $S(8,3)$ is the Stirling number of second kind.
                Recall that the condition each department has to receive at least one employee means that each map has to be surjective.




                The number of surjective maps from an $n$ elements set onto a $k$ elements set is given by:
                $$S(n,k)cdot k!$$







                share|cite|improve this answer

























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  This is equivalent to count the number of surjective maps from the set of employees $e_1,dots,e_8$ to the set of departments $d_1,d_2,d_3$. So you get
                  $$S(8,3)times 3!=5796quad textways$$
                  where $S(8,3)$ is the Stirling number of second kind.
                  Recall that the condition each department has to receive at least one employee means that each map has to be surjective.




                  The number of surjective maps from an $n$ elements set onto a $k$ elements set is given by:
                  $$S(n,k)cdot k!$$







                  share|cite|improve this answer















                  This is equivalent to count the number of surjective maps from the set of employees $e_1,dots,e_8$ to the set of departments $d_1,d_2,d_3$. So you get
                  $$S(8,3)times 3!=5796quad textways$$
                  where $S(8,3)$ is the Stirling number of second kind.
                  Recall that the condition each department has to receive at least one employee means that each map has to be surjective.




                  The number of surjective maps from an $n$ elements set onto a $k$ elements set is given by:
                  $$S(n,k)cdot k!$$








                  share|cite|improve this answer















                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Jul 31 at 4:03


























                  answered Jul 31 at 3:20









                  Hector Blandin

                  1,741716




                  1,741716






















                       

                      draft saved


                      draft discarded


























                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2867603%2fhow-to-apply-the-principle-of-inclusion-and-exclusion-in-this-problem%23new-answer', 'question_page');

                      );

                      Post as a guest
































































                      Comments

                      Post a Comment

                      Popular posts from this blog

                      What is the equation of a 3D cone with generalised tilt?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                      Read more

                      Color the edges and diagonals of a regular polygon

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                      Read more

                      Relationship between determinant of matrix and determinant of adjoint?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                      Read more