Question 941398
What is the number of ways letters in the word KOMBINATOORIKA can be rearranged, 
<pre>
Here's my take on it, but I didn't get 710579520, though:

In alphabetical order, the letters are

AABIIKKMNOOORT

Start with the 5 letters which occur only once in AABIIKKMNOOORT,
which are B,M,N,R,T

There are 5! or 120 permutations of BMNRT. In each put a space between each two,
a space before the first, and a space after the last.  Like this:

__B__M__N__R__T__

[We will do this after each time e choose spaces to put letters in.] 

The above is a good representative of those 120 with the 6 spaces: 

__B__M__N__R__T__

For each of the 120 arrangments like that, we can pick 2 of the
6 blanks to put the 2 A's --- in 6C2 = 15 ways. That's 120*15 = 1800.

A good representative of these 1800 is

__A__B__M__N__A__R__T__

For each of the 1800 arrangments like that, we can pick 2 of the
8 blanks to put the 2 I's --- in 8C2 = 28 ways. That's 1800*28 = 50400.

A good representative of these 50400 is

__A__B__M__I__N__A__R__T__I__

For each of the 50400 arrangments like that, we can pick 2 of the
10 blanks to put the 2 K's --- in 10C2 = 45 ways. That's 50400*45 = 2268000.

A good representative of these 2268000 is

__A__B__M__K__I__N__K__A__R__T__I__

For each of the 2268000 arrangments like that, we can pick 3 of the
12 blanks to put the 3 O's --- in 12C3 = 220 ways. That's 2268000*220 = 498960000.

A good representative of these 498960000 is

__O__A__B__O__M__K__I__K__O__N__A__R__T__I__

Now we can drop the spaces and no two adjacent letters are the same.

OABOMKIKONARTI

My answer: 498960000

Edwin</pre>