Win the Coin Toss Game at the Fair - Challenge Problem

[Ann1]

Marci is working a coin toss game at the fair. You get 4 coins to toss to try to get numbers that add up to 10. You can get a
0,1,2,3,4 or 5. How many different combinations of 10 are there? (Hint: 5+5+0+0 is different from 5+0+5+0)

 

[aditup]

Combinations

Marci is working a coin toss game at the fair. You get 4 coins to toss to try to get numbers that add up to 10. You can get a 0,1,2,3,4 or 5, How many different combinations of 10 are there? (Hint 5+5+0+0 is different than 5+0+5+0)

 

[MarkFL]

Hello and welcome to MHB, Ann and aditup! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[aditup]

I started with the combination of how many different ways you can add 4 coins to get 10.

5+5+0+0
5+4+1+0
5+3+2+0
5+2+2+1
4+4+2+0
4+3+2+1
4+3+3+0
3+3+3+1
3+2+3+2

 

[MarkFL]

I started with the combination of how many different ways you can add 4 coins to get 10.

5+5+0+0
5+4+1+0
5+3+2+0
5+2+2+1
4+4+2+0
4+3+2+1
4+3+3+0
3+3+3+1
3+2+3+2

Okay...good! Now, let's first look at those combinations having 4 distinct values:

5+4+1+0
5+3+2+0
4+3+2+1

How many ways can we arrange each?

 

[IHateFactorial]

Marci is working a coin toss game at the fair. You get 4 coins to toss to try to get numbers that add up to 10. You can get a
0,1,2,3,4 or 5. How many different combinations of 10 are there? (Hint: 5+5+0+0 is different from 5+0+5+0)

Well... There are plenty of ways to get 10 by adding 4 numbers from 0-5, let's see them:

0055 1315 2224
0145 1225 2233
0235 1144 2242
0244 1234
0334 1333
(continuing would simply mean repeating the same addition, so we're not going to do that.

I BELIEVE that's all of the UNIQUE ways of doing it.

So there are 13 unique ways, each with 4! ways of arranging them, so it'd be \(\displaystyle 4! \times\ 13\).

 

[IHateFactorial]

Well... There are plenty of ways to get 10 by adding 4 numbers from 0-5, let's see them:

0055 1315 2224
0145 1225 2233
0235 1144 2242
0244 1234
0334 1333
(continuing would simply mean repeating the same addition, so we're not going to do that.

I BELIEVE that's all of the UNIQUE ways of doing it.

So there are 13 unique ways, each with 4! ways of arranging them, so it'd be \(\displaystyle 4! \times\ 13\).

Sorry, my bad. I forgot to take into account the repeated numbers in each unique combination.

I.e. 1333 can only be cominated in 4 ways, no 4! ways (they'd be: 1333, 3133, 3313, 3331).

I'll fix that in a bit.

Factoring in for repeated numbers we have
\(\displaystyle (4 \cdot 4!) + (4 \cdot 4!/2!) + (3 \cdot 4!/2!2!) + (2 \cdot 4!/3!) = 96 + 48 + 18 + 8 = 200\)
ways of getting 10.