ENTERTAINMENT
In alphabetical order, the letters are A,E,E,E,I,M,N,N,N,R,T,T,T.
First we find the number of distinguishable arrangements, regardless of
whether the three E's are apart or not.
That's 13 things with 3 indistinguishable E's, 3 indistinguishable N's, and 3
indistinguishable T's,
That's 
From that we will subtract the ones with some E's together. They consist of
two cases:
Case 1. the number of arrangements that have all three E's together, like EEE.
That's the number of arrangements of these 11 "things" with 3
indistinguishable N's and 3 indistinguishable T's.
A,EEE,I,M,N,N,N,R,T,T,T.
That's 
ways.
and
Case 2. the number of arrangements that have EE together and the E apart from it.
First we find the number of arrangements without any E's. That's the number
of arrangements of these 10 things, which have 3 indistinguishable N's and 3
indistinguishable T's:
A,I,M,N,N,N,R,T,T,T.
That's 
Then we'll insert an EE and an E among them so that we don't put them
together, avoiding counting again the ones with EEE from case 1.
To do that, we now look at a random arrangement from the 100800 with no E's at
all, say this one:
T,R,N,A,I,T,N,N,M,T
We put 8 spaces between the letters, 1 space in the beginning, and 1 space at
the end. That's 10 spaces, and we'll put an E in one of them and EE in another
one. That way the E and the EE won't be together.
_T_R_N_A_I_N_N_M_T_
We can choose the space to put the single E in in 10 ways.
That leaves 9 spaces to put the double EE in.
So for each of the 100800 ways we can insert the E and the EE in 10∙9 or 90
ways. That's 100800∙90 = 9072000 ways to have EE separate from E.
So the total number that we must subtract from 28828800 is 1108800 from case 1
and 9072000 from case 2.
28828800-1108800-9072000 = 15348000 ways.
Edwin
