The number of different four-letter arrangements (meaningful or not) that can be formed from the letters of the word MATHEMATICS is

The letters are $M,M,A,A,T,T,H,E,I,C,S$: three repeated pairs and five single letters, i.e. $8$ distinct letters. Case 1, all four distinct: $\binom84\cdot4!=1680$. Case 2, exactly one pair plus two different other letters: choose the pair in $3$ ways, the other two from the remaining $7$ distinct letters in $\binom72=21$ ways, and arrange $\dfrac{4!}{2!}=12$ ways, giving $3\cdot21\cdot12=756$. Case 3, two pairs: $\binom32=3$ choices, each arranged in $\dfrac{4!}{2!2!}=6$ ways, giving $18$. The total is $1680+756+18=2454$. The value $1680$ ignores the repeated letters.