MATHEMATICS

There are 2 indistinguishable M's, 
2 indistinguishable A's and 
2 indistinguishable T's.

Case 1: arrangements of 4 distinguishable
letters, such as TIME and HEAT.

There are 8 distinguishable letters from
this set: {M,A,T,H,E,I,C,S}

Choose the 1st letter 8 ways.
Choose the 2nd letter 7 ways.
Choose the 3rd letter 6 ways.
Choose the 4th letter 5 ways.

That's 8*7*6*5 = 8P4 = 1680 arrangements
for case 1.

Case 2: arrangements of exactly one pair of
indistinguishable letters, such as MASS or THAT.

Choose the letter for the pair of
indistinguishable letters 3 ways,
from (M,A,T}
Choose the 2 positions in the arrangement
from the set of positions:

{1st letter, 2nd letter, 3rd letter, 4th
letter} 

for the pair of indistinguishable letters to
go 4C2 = 6 ways.
Choose the letter for the left-most
unfilled position 7 ways.
Choose the letter for the rightmost-most
unfilled position 6 ways.

That's 3*6*7*6 = 756 ways.

Case 3: arrangement of two pair of 
distinguishable letters, such as MAMA or TATA.

Choose the two letters from {M,A,T} for the
pairs of indistinguishable.
That's 3C2 = 3 ways.
Choose the positions for the pair in 4C2=6
ways.

That's 3*6 or 18 ways.

Grand total: 1680+756+18 = 2454 
distinguishable 4-letter arrangements from
MATHEMATICS
 
Edwin