7 non-repeating (distinct) letters on a scrabble rack can be arranged in 5,040 ways.
For example, ABCDEFG = 7! arrangements. Each arrangement represents one unique word.

A blank tile can represent any single letter A to Z so that if we have 6 distinct letters + 1 blank tile:
ABCDEF? we get 115,920 unique words.

Using brute-force word generation I got these word counts for the following 7 racks:
?=26 A?=51 AB?=150 ABC?=588 ABCD?=2,880 ABCDE?=16,920 ABCDEF?=115,920

Case 1
I wanted to know the formula that calculates the unique word count w for the 7
racks above in terms of one blank tile b and (n-b) distinct letters. Rephrasing,
how many unique words w can we make if we have a rack with 0 or more distinct letters + 1 blank tile.

Solution:
n varies from 1 to 7, b = 1
One blank tile can be any one of the 26 alphabet letters,
n = number of distinct letters + 1 blank
b = number of blanks
w = number of distinct words

		/	(n - b)	\
w =	n!	(26 -	--------	) formula 1
		\	(b+1)!	/

Example:

		/	(7 - 1)	\
w =	7!	(26 -	--------	) = 115,920 unique words
		\	(2)!	/

Formula 1 gives the correct word count for each of the 7 racks above.

	rack	w
n=1 b=1	?	26
n=2 b=1:	A?	51
n=3 b=1:	AB?	150
n=4 b=1:	ABC?	588
n=5 b=1:	ABCD?	2,880
n=6 b=1:	ABCDE?	16,920
n=7 b=1:	ABCDEF?	115,920

Case 2
Back to brute force unique word generation I got these results for racks with 2 blank tiles:

	rack	w	expected w using formula 1
n=2 b=2:	??	676	52 WRONG
n=3 b=2:	A??	1,951	155 WRONG
n=4 b=2:	AB??	7,502	616 WRONG
n=5 b=2:	ABC??	36,030	3,060 WRONG
n=6 b=2:	ABCD??	207,480	18,240 WRONG
n=7 b=2:	ABCDE??	Out of memory	126,840 WRONG

Obviously formula 1 does not work when we have 2 blank tiles and it looks like I'll
never know w for the last rack sequence. Formula 1 appears to be a special case
for a more general formula where b can be 0, 1 or 2

1st question: What is the general formula to calculate the w unique words
if we have a rack with 0 or more distinct letters + (0, 1 or 2 blank tiles)?

A blank tile can represent any single letter A to Z.
Distinct letters means a non repeating sequence like ABC not AAA, AAB, AAC ...CCC
Minimum rack length is 1 tile max is 7 tiles.

w = F(n,b) for any of these 20 racks:
where n = total number of tiles 1..7
b = total number of blank tiles 0..2
w = total unique words

n=1:	A	?	-
n=2:	AB	A?	??
n=3:	ABC	AB?	A??
n=4:	ABCD	ABC?	AB??
n=5:	ABCDE	ABCD?	ABC??
n=6:	ABCDEF	ABCDE?	ABCD??
n=7:	ABCDEFG	ABCDEF?	ABCDE??

2nd question: What is the complete general formula when the lettered tiles
may repeat and may be combined with 0, 1 or 2 blank tiles. 

If any of the lettered (non-blank) tiles do repeat then we must
divide n! by the product of factorial of the respective repeating letters.
ABBCCCD: w = n!/(1! 2! 3! 1!) = 5040/12 = 420

But that's as far as I can get.

Your assistance would be greatly appreciated.

Thanks

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