How many distinct four-letter permutations are possible using four of the ten letters in MATHCOUNTS?
There are (410)=210 ways to choose which 4 letters to use out of the 10 letters in MATHCOUNTS.
Once we have chosen our 4 letters, we can permute them in 4! ways.
Therefore, there are 210×4!=5040 distinct four-letter permutations that can be formed using four of the ten letters in MATHCOUNTS.