Probability of drawing 6 specific letters from a set of 144 with 21 draws 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

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: 
$${1996005899239917869870709\over 8943919960348862933558400}\approx .22317.$$
