# Thread: Possibilities in a given set

1. ## Possibilities in a given set

There are 3 men A,B and C. They have 4 sons in total.
What are the total possibilities of the order of no. of sons they have.
How do we find it.
Plz help I am totally new to this topic.

2. ## Re: Possibilities in a given set

Originally Posted by AaPa
There are 3 men A,B and C. They have 4 sons in total.
What are the total possibilities of the order of no. of sons they have.
How do we find it.
Solve $\displaystyle A+B+C=4$ in the non-negative integers.
We can put N identical objects into K different cells in
$\displaystyle \binom{N+K-1}{N}$ ways.
Here which is which?

3. ## Re: Possibilities in a given set

Originally Posted by Plato
$\displaystyle \binom{N+K-1}{N}$ ways.
I don't understand this. I am not being taught this anywhere.
It just came to my mind randomly.

4. ## Re: Possibilities in a given set

Originally Posted by AaPa
I don't understand this. I am not being taught this anywhere.
It just came to my mind randomly.
Well then make a list:
400
310
301
220
etc

5. ## Re: Possibilities in a given set

But I wanted to do it using a formula.

7. ## Re: Possibilities in a given set

thanks a lot.

8. ## Re: Possibilities in a given set

N=4 sons (1234) to separate into K=3 groups, using K-1=2 dividers (--).
So 123-4- means 3,1,0
So 1-23-4 means 1,2,1
So we have to choose K-1=2 dividers among N+K-1=4+2=6 symbols,
so #ways is (6 over 2), or (4+3-1 over 2), or (N+K-1 over K-1), or (N+K-1 over N)

9. ## Re: Possibilities in a given set

Ya; and to "see them" as Plato suggested, just write 'em down in ascending order:
004
013
022
031
040....5
103
112
121
130....4
202
211
220....3
301
310....2
400....1
.........15