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Dora and Her Game 
    of Go Fish




                     1
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




     Okay first lets see what kind of question this is. 
     This is a simple combination's problem. How 
     many ways can we arrange 3 things or in this 
     case 3 people.




                                                                     2
We can find this by solving for 3 factorial (3!). No, the exclamation mark does 
not mean the number 3 said louder. It means the number 3 multiplied by (3­1) 
then (3 ­ 2) and so on until we reach a difference of one.




                                                                                   3
Okay so the factorial sign (!) means that we multiply 3 with 
(3 ­ 1) and (3 ­ 2) which is 3 x 2 x 1.




                                                                4
So the product of 3 x 2 x 1=6. So therefore there are 6 
ways that Dora, Boots, and Benny can sit together.




                                                           5
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




    Okay so we got that little trio trouble over with. Since we 
    put them in a bag, as in they are counted as one whole 
    object now since they can't be separated. Our next 
    objective is to find how many ways those 4 things can be 
    arranged. 




                                                                     6
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




    But wait, this is not just an arrangement of people it is an 
    arrangement of people around a circular table. This means 
    there has to be a reference person. This has to be done 
    because a circle doesn't have a beginning nor an end.




                                                                     7
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




    Remember that the formula that is to be used is (n­1)!. 
    Normally it would just be n! but since a circle doesn't have 
    an end point, we use a person to be that point resulting in 
    one less of a person. Okay now we know enough to answer  
    this part of the question.




                                                                     8
There are 4 objects but in this case 4 people. But 
remember there has to be 1 person to be used as a 
reference point. Then the others can be arranged in 
any order.




                                                       9
This then makes 3!. Which means 3 x 2 x 1. Just like in 
the 1st part of the problem.




                                                           10
So therefore the number of ways 4 people 
can be seated around a round stump is 6 
ways.




                                            11
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




        Since we know how many ways we can arrange Dora' Trio 
        and the 4 objects around the stump we can find how many 
        ways these things can be arranged in total. All we have to 
        do is add the number of ways that Dora's Trio can be 
        arranged by themselves to the number of ways 4 people 
        can be arranged around a round table.




                                                                      12
So then we just add the values that we got earlier and we get 
12. But wait how about the second obstacle. Diego doesn't 
want to sit beside Dora because he doesn't want a cheater 
around.




                                                                 13
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




       Okay since Diego doesn't want to sit beside Dora we have to get 
       rid of those combinations. Well there are only a few ways that 
       they could be beside each other. It can only happen whenever 
       Dora is sitting on the side of the Trio. Plus if that happens Diego 
       only has one spot that he could sit that would cause them to sit 
       beside each other.




                                                                              14
How many ways can Dora and her friends sit at the tree stump to 
play cards when Dora has to sit with Boots and Benny? In addition 
Diego doesn't want to sit with Dora.




         So therefore then all we have to do is get rid of those 
         combinations. The total of combinations that we solved earlier 
         subtract 2.


                        12 ­ 2 = 10
         Then that means there are 10 ways that we can arrange 
         Dora's Go Fish Shindig on a circular table with those 
         proposals.




                                                                           15

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Dora and Her Game of Go Fish Part 1