1.
Dora and Her Game
of Go Fish
1

2. 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

3. 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

4. Okay so the factorial sign (!) means that we multiply 3 with
(3 ­ 1) and (3 ­ 2) which is 3 x 2 x 1.
4

5. 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

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.
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

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.
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

8. 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

9. 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

10. This then makes 3!. Which means 3 x 2 x 1. Just like in
the 1st part of the problem.
10

11. So therefore the number of ways 4 people
can be seated around a round stump is 6
ways.
11

12. 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

13. 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

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.
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

15. 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.
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