Possible combinations

[opala]

Hi guys,
I have a math problem I'm not sure how to solve.

I have X (lets say 11) Marbles and a box, each day up to Y (lets say 35) days I can put any amount of marbles in the box, but by the end of the 35 days I must be left with 0 marbles.

for example,
I can put 11 marbles on day 1 and then 0 the rest of the time.
I can put 1 marble each day up until the 11th day then 0 the rest of the month.
I can put 1 on the first day and 10 on the last day.
I can put 1 marble every 3 days etc...

the question is, how many possible different combinations are there? I'm not sure how to calculate this at all, the final number is probably in the millions isnt it?

thanks

 

[tkhunny]

Ever hear of "Stars and Bars"? Here's a useful video:

Keep in mind that he does not allow two bars in the same slot (representing a zero (0)). You will have to allow that.

Let's see what you get.

 

[pka]

I have X (lets say 11) Marbles and a box, each day up to Y (lets say 35) days I can put any amount of marbles in the box, but by the end of the 35 days I must be left with 0 marbles.
for example,
I can put 11 marbles on day 1 and then 0 the rest of the time.
I can put 1 marble each day up until the 11th day then 0 the rest of the month.
I can put 1 on the first day and 10 on the last day.
I can put 1 marble every 3 days etc...
the question is, how many possible different combinations are there? I'm not sure how to calculate this at all, the final number is probably in the millions isnt ?

The general rule for placing \(\displaystyle N\) identical objects into \(\displaystyle K\) distinct cells is \(\displaystyle \dbinom{N+K-1}{N}=\frac{(N+K-1)!}{(N!)(K-1)!}\)
The video above give the rational for that formula.

 

[opala]

The general rule for placing \(\displaystyle N\) identical objects into \(\displaystyle K\) distinct cells is \(\displaystyle \dbinom{N+K-1}{N}=\frac{(N+K-1)!}{(N!)(K-1)!}\)
The video above give the rational for that formula.

Hi, thanks for the reply, I've learned about stars and bars now

so plugging in the numbers into the formula (11 identical objects ("marbles") into 35 cells ("days"), even though its 1 box and 35 days I am assuming in this case its the same, this results in over 10 billion possibilities, or am I doing something wrong? that seems pretty crazy

thanks