You can
put this solution on YOUR website! I will assume the two T's are indistinguishable and
that there can be two T's in the same word.
The number of words that have all different letters is
6P1 + 6P2 + 6P3 + 6P4 + 6P5 + 6P6 =
6 + 30 + 120 + 360 + 720 + 720 = 1956
In addition to that number, we must calculate:
A. The number of 2 letter words with 2 T's
There's only 1, which is TT.
B. The number of 3 letter distinguishable words with 2 T's:
There are 3C2 = 3 positions to place the 2 T's.
There are 5P1 = 5 ways to place the 1 non-T.
That's 3*5 = 15 ways.
C. The number of 4 letter distinguishable words with 2 T's:
There are 4C2 = 6 positions to place the 2 T's.
There are 5P2 = 20 ways to place the 2 non-T's.
That's 6*20 = 120 ways.
D. The number of 5 letter distinguishable words with 2 T's:
There are 5C2 = 10 positions to place the 2 T's.
There are 5P3 = 60 ways to place the 3 non-T's.
That's 10*60 = 600 ways.
E. The number of 6 letter distinguishable words with 2 T's:
There are 6C2 = 15 positions to place the 2 T's.
There are 5P4 = 120 ways to place the 4 non-T's.
That's 15*120 = 1800 ways.
F. The number of 7 letter distinguishable words with 2 T's:
There are 7C2 = 21 positions to place the 2 T's.
There are 5P5 = 120 ways to place the 5 non-T's.
That's 21*120 = 2520 ways.
[As a check on that last number of ways, we can use the usual
formula for the number of distinguishable words that can be
formed from a 7-letter word with 2 indistinguishable letters,
7!/2! = 5040/2 = 2520]
The total number of words with two T's is
1 + 15 + 120 + 600 + 1800 + 2520 = 5056
Grand total = 1956 + 5056 = 7012
Edwin
