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I know similar questions have been asked before, but I am struggling to find a method to calculate the probability for the situation below:

I have a set of 144 scrabble tiles with the following letter distribution.

2 x , K, Q, X, Z

3 x B, C, F, H, M, P, V, W, Y

4 x G

5 x L

6 x D, S, U

8 x N

9 x T, R

11 x O

12 x I

13 x A

18 x E

I am trying to calculate the probability having the letters "I S N E A T" in any order when 21 tiles are picked from the set of 144.

I would appreciate any help on methods to calculate this

Thanks in advance - Gary

gary
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1 Answers1

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There are a total of ${144\choose 21}=8943919960348862933558400$ ways to choose 21 tiles from the 144 available.

Let $I=\{12,6,8,18,13,9\}$ be the counts of tiles corresponding to the letters I,S,N,E,A,T. Then the number of bad samples, i.e, missing one of the required letters, of size 21 can be worked out using the inclusion-exclusion principle, giving $$\sum_{A\subseteq I, A\neq\emptyset} {144-S(A)\choose 21}(-1)^{|A|+1} =6947914061108945063687691. $$ Here $S(A)=\sum_{x\in A}x$ gives the number of tiles corresponding to the set $A$. For instance, when $A=\{12,6\}$ we have $S(A)=18$, since there are eighteen tiles labelled I or S.

Subtracting gives the number of good samples $1996005899239917869870709$.

Dividing by the total number of samples gives the chance of getting all six letters I,S,N,E,A,T: $${1996005899239917869870709\over 8943919960348862933558400}\approx .22317.$$