How many possible Scrabble racks are there at the beginning of the game? In the American version of Scrabble version there are how many possible starting racks.  
A rack contains exactly 7 letters (blanks count as a letter for this problem).  Most of the letters in scrabble are constrained meaning that a rack can contain: 
At most 2 of  B, C, M, P, F, H, V, W, Y, or blanks
At most 1 of   K, J, X, Q, Z, 
AT most   3 G, 4 D, 4 U, 4 S, 4 L 6 T, 6 R, 6 N, 7 A, 7 I, 7 E, 7 O, 7 U 
I guess this would be the number of positive integer solutions for
the equation ..................................
a +b +c +d +  ... +x +y +z +blanks = 7. 
Where a is greater than or equal to 0 and less than or equal to 7, 
  b is greater than or equal to zero and less than or equal to 3 ....

Edited to use the correct letter distributions, as pointed out by Peter.
There are:


*

*4 letters with 7 tiles;

*3 letters with 6 tiles;

*4 letters with 4 tiles;

*1 letter with three tiles;

*10 letters with 2 tiles; and

*5 letters with 1 tile.


So the answer is the coefficient of $x^7$ in:
$(x^7+x^6+x^5+x^4+x^3+x^2+x+1)^4 \times (x^6+x^5+x^4+x^3+x^2+x+1)^3 \times$
$(x^4+x^3+x^2+x+1)^4 \times (x^3+x^2+x+1) \times (x^2+x+1)^{10} \times (x+1)^5$
According to Wolfram Alpha, this is 3199724.
