how many combinations of tents are possible to sleep 28 if all tents are fully occupied and exactly one 12-peerson tent is used?

Question

I need help finding this answer.

 

The tents hold 2, 3, 5, 6 or 12 people.

Answer 1

For any Word (Story) question, read and re-read it until you understand it and can put it into your own words.  This problem statements includes a constraint:  "exactly one 12-person tent is used."  That means the problem really is, "How many combinations of fully-occupied tents (of 2, 3, 5, or 6 people) are needed for 16 people?"

 

Now, remember the decimal positional number system:

      The number:   abc is

                   a               b                c

                0-9              0-9             0-9

   There are 10*10*10=1000 possibilities ranging from 000 to 999.

 

So, let's create a "tent" number system:

 

                        Holds 6     Holds 5     Holds 3     Holds 2

   No. of tents:       0-2         0-3           0-5          0-8    (total people must be 16)

 

It's easier to understand counting down:

  Combination #    2-0           3-0             5-0            8-0

          -------          ------         -----           -----           ----

             1                 2               0               0               2
             2                 1               2               0               0
             3                 1               1               1               1
             4                 1               0               2               2
             5                 1               0               0               5
             6                 0               2               2               0
             7                 0               2               0               3
             8                 0               1               3               1
             9                 0               1               1               4
            10                0               0               4               2
            11                0               0               2               5
            12                0               0               0               8

 

There are 12 combinations that can house 16 people.

 

Note:  This is called "exhaustive enumeration" because we considered every possibility.

 

FOR A GOES FROM 2 TO 0 BY -1
   FOR B GOES FROM 3 TO 0 BY -1
      FOR C GOES FROM 5 to 0 by -1
         FOR D GOES FROM 8 TO 0 BY -1
             IF (A*6+B*5+C*3+D*2 = 16) THEN OUTPUT A,B,C,D
         NEXT D
      NEXT C
   NEXT B

Answer 2

 

You already know you're going to put 12 people in one tent, and can't use a 12 person tent again. So you have 16 people left to put into tents. 

 

You're left thinking of combinations of 2, 3, 5, and 6 that add to 16. 

 

Come up with an organized way to figure this out. The way I think to do it, is let's see how many times 6 goes into 16 and try to fill in the gap left with the next largest numbers.

 

6, 6, 2, 2      (I put 6 in twice which left four to get to 16, so only twos worked)

 

6, 5, 5          (I went with 6 once, fives were the next biggest, so I did that. Only one way.)

 

6, 5, 3, 2      (Doing 5 only once, there is only one way.)

 

6, 3, 3, 2, 2  (Can't choose 3 threes. Only one way to choose two threes.)

 

6, 2, 2, 2, 2, 2        (Can't choose only one three with what's left (unless I want to repeat what I have above) so this is the last option for one six.)

 

Let's do 5s. If I pick 3 fives, then I'm left needing a tent for only one person. So I'll start by picking two:

 

5, 5, 3, 3                (Can't use three and two fives after this, so...)

 

5, 5, 2, 2, 2             (This is the last option for two fives.)

 

5, 3, 3, 3, 2             (Put as many threes as possible and then plugged up the hole.)

 

5, 3, 2, 2, 2, 2         (Can't do 5 and two 3s because you can't add twos to an odd number and get an even number.)

 

And that is it, because if I add any number of 2s to 5, I'll get an odd number, not 16.

 

3, 3, 3, 3, 2, 2         (Can't do 5 threes because that's 15 and no 1 person tent...)

 

3, 3, 2, 2, 2, 2, 2      (Can't do 3 threes because that's 9 and adding more twos will only get me odd numbers.)

 

Lastly:

 

2, 2, 2, 2, 2, 2, 2, 2

 

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There might be a much better way to do this.