The seven letters of the word BALLOON are arranged at random in a row. What is the probability that the two L's do NOT stand next to each other?

Count places rather than whole words. The two L's take up 2 of the 7 places and each of the C(7,2) = 21 pairs of places is equally likely; exactly 6 pairs are neighbours — (1,2), (2,3), … , (6,7) — so P(the L's are together) = 6/21 = 2/7 and P(they are apart) = 1 − 2/7 = 5/7. The long route agrees: BALLOON has 7!/(2!·2!) = 1260 distinct arrangements, of which 6!/2! = 360 tie the two L's into one block, and (1260 − 360)/1260 = 5/7. Answering 2/7 solves the opposite event, while forgetting that the two O's are alike inside the block count gives 720 together, hence the wrong pair 4/7 and 3/7.