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I would like to compute the number of Scrabble draws, when starting a game. Translated to mathematics, this asks for the number of sub-multisets of a multiset.

Lets say we have a multiset $M$ over a set $N=\{1,...,n\}$. This means there are numbers $m_1,..m_n$, which indicate the multiplicity of the numbers $1$ to $n$. For $k \leq m := m_1+...+m_n$ we now want to draw a $k-$multiset from $M$, in other words numbers $k_1,...,k_n$ with the property $k_1+...+k_n=k$ and $0 \leq k_i \leq m_i$.

My Ansatz was drawing a number of $1$'s, this is $k_1$. In order to do so, we have $\min(m_1, 1+k)$ many possibilities.

Then we draw $k_2$, the number of $2$'s. For this, we have $\min(m_2, 1+k - k_1)$ many possibilities.

...

Finally, for $k_n$, there are $\min(m_n, 1+k- (k_1 + ... +k_{n-1}))$ many possibilities.

The problem is the obviously the self referentiality, i.e the number of possibilities for $k_i$ depending on $k_j$ for $j < i$. I'm sure there is a very clean way to do it via binomial coefficients or so but I'm not sure how to do it that way. Any help is appreciated. :-)

EDIT: bump

Arturo Magidin
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durst
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2 Answers2

2

I don't know any tricks to make this easier.

I got the number of tiles of each letter and the blanks (2) from here and then counted the number of ways with the R program below. For example, you can draw 0 "A"s and 7 tiles from the remainder, or 1 "A" and 6 tiles from the remainder, or 2 "A"s and 5 tiles from the remainder,... The total can be counted recursively. I got a total of 3,199,724 different ways.
This is related.

x=sort(c(9, 2, 2, 4, 12, 2, 3, 2, 9, 1, 1, 4, 2, 6, 8, 2, 1, 6, 4, 6, 4, 2, 2, 1, 2, 1,2))

f=function(x,k) {
  if (sum(x)<k) cnt=0 else if (sum(x)==k | (length(x)==1)) cnt=1 else {
    cnt=0
    for (i in 0:min(x[1],k)) cnt=cnt+f(x[-1],k-i)
  }
  return(cnt)
}
f(x,7)
John L
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For a general case, without specific values for $m_1, \ldots m_n$, see here and/or here, whereas, if we have $m_1=m_2=\ldots=m_n$ there are simpler formulas here or here ($k_i=0$ not allowed there).

If we have specific numeric values for $m_1, \ldots m_n$, then we use generating functions more easily to get the number of $k$-multisets as the coefficient of $x^k$ of the generating function:

$$f(x) = \prod_{i=1}^n (1+x+\ldots+x^{m_i}) = \prod_{i=1}^n \frac{x^{m_i+1}-1}{x-1}$$ which in our case is (using Wikipedia data, the same as @John L's answer):

$$f(x)=\frac{(x^2-1)^5(x^3-1)^{10}(x^4-1)(x^5-1)^4(x^7-1)^3(x^9-1)(x^{10}-1)^2(x^{13}-1)}{(x-1)^{27}}$$

And using Wolfram Alpha get some coefficients, e.g.:

$$[x^7]f(x)=3199724$$

like in @John L answer.

Note also that the total number of multisets of any size is:

$$\prod_{i=1}^n (1+m_i)$$

Fabius Wiesner
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