If the like letters of CINCINNATI were subscripted differently, 
like this:

C1I1N1C2I2N2N3ATI3

There would be 10! distinguishable permutations.

If we take the subscripts off the C's, there will be 2! 
exact duplicates among the 10!

So we must divide the 10! by 2! and get  distinguishable
permutations of

CI1N1CI2N2N3ATI3 

If we take the subscripts off the I's, there will be 3! 
exact duplicates among the 

So we must divide the  by 3! and get  
distinguishable permutations of

CIN1CIN2N3ATI


If we take the subscripts off the N's, there will be 3! 
exact duplicates among the 

So we must divide the  by 3! and get  
distinguishable permutations of

CINCINNATI

Answer:  =  =  = 50400.

Edwin