The seven letters of the word CRICKET are rearranged. In how many of the resulting strings do the two C's fail to be neighbours?

CRICKET has 7 letters with C repeated twice, so the unrestricted total is 7!/2! = 2520. For the two C's together, glue them into one block: the units [CC], R, I, K, E, T are six distinct items giving 6! = 720. Subtract: 2520 − 720 = 1800. Trap 720 answers the opposite event, 'always together'. Trap 2520 ignores the restriction altogether. Trap $1440=2\times 720$ multiplies the block by 2! for an internal swap, but the two C's are identical, so swapping them changes nothing.