The nine letters of the word COMMITTEE are rearranged. In how many of the resulting letter-strings do all four vowels stand together as one unbroken group?

COMMITTEE splits into consonants C, M, M, T, T and vowels O, I, E, E. Glue the four vowels into one block: the units to arrange are [block], C, M, M, T, T — six units in which M and T each repeat, giving 6!/(2! × 2!) = 720/4 = 180. Inside the block the four vowels rearrange in 4!/2! = 12 ways, because E appears twice. Total = $180\times 12=2160$. Trap $4320=180\times 24$ shuffles the block as though its two E's were distinguishable. Trap 45360 = 9!/(2! × 2! × 2!) is the unrestricted total. Trap 1080 slips in an extra division by 2.