The nine letters of the word INSTITUTE are rearranged. In how many of the resulting letter-strings do the three T's stand together as one unbroken block?

INSTITUTE has 9 letters: T three times, I twice, and N, S, U, E once each. Glue the three T's into a single block. The units to arrange are then [TTT], I, N, S, I, U and E — seven units in which I repeats, so 7!/2! = 5040/2 = 2520. Trap $15120=2520\times 3$! shuffles the three T's inside the block, but they are identical, so swapping them produces nothing new. Trap 5040 = 7! forgets that the letter I appears twice among those seven units. Trap 30240 = 9!/(3! × 2!) is the unrestricted total with no block at all.