0

I recently came across this problem and couldn't even start on it. Would someone be able to help me?

Given $m$ identical symbols, say H's, show that the number of ways you can distribute them in $k$ boxes, marked $1,2, \ldots, k$ so that no box goes empty is $\dbinom{m-1}{k-1}$.

Thanks in advance.

Michael Hardy
  • 1
coolcheetah
  • 349

2 Answers2

2

We can use a classic combinatorial tactic called stars and bars. Because it is conventional, we will represent each symbol as $\star$ instead of H. Consider $m=6$:

$$\star \star \star \star \star \star$$

We can then place bars $\vert$ to represent the edges separating boxes. Say $k=3$. Then for example we can have:

$$\star \vert \star \star \star \vert \star \star$$

This represents the case where there is one star in the first box, three stars in the second box, and two stars in the third box. In general, we can place at most one bar in any gap between two stars. For $m$ stars, there will be $m-1$ gaps, and we will need to choose $k-1$ of them to place bars in if we want to define $k$ boxes. Hence, there are in total ${m-1}\choose{k-1}$ possible arrangements.

hexaflexagonal
  • 1,426
  • Thank you. I get it now. Doesn't this technique have utility in finding the number of partitions of a number (ie a + b + c + d = 100) also? – coolcheetah Jul 24 '15 at 17:51
  • Good observation! I presume you mean that we could write $1+1+1+\dots+1=m$ and then "box" $k$ groups of addition. If you do not allow for members of the partition to be $0$, then you can use the method above. If you do, then you can use theorem two on the Wiki page. But beware: if we are separating $m$ into, say, $k=2$ summands (call then $a$ and $b$), these methods count as distinct the case where $a=x,b=y$ and the case where $a=y,b=x$. – hexaflexagonal Jul 24 '15 at 18:01
  • That's what I meant. Thanks for the quick reply! Also, do you have any tips for me to study combinatorics? For whatever reason, even though it is kinda intuitive I end up using wrong techniques in combinatorics problems. Is there a way you can classify problems and use certain techniques and methods in certain problems? – coolcheetah Jul 24 '15 at 18:05
  • 1
    I'm not sure how much help I can be there, but here is a reference request thread for introductory combinatorics books. – hexaflexagonal Jul 24 '15 at 18:12
1

hint

  1. Begin by putting one symbol in each box. That leaves $m-k$ to be distributed.
  2. Distributing $m-k$ among $k$ boxes can be done in how many ways? (see stars and bars problem)
  3. first step can be done in 1 way, how many ways total for the entire procedure?
gt6989b
  • 54,422