The eleven letters of the word ENGINEERING are shuffled and laid out again as a single string. How many different strings are possible?

ENGINEERING has 11 letters made up of E three times, N three times, G twice, I twice and R once. Distinct arrangements = 11!/(3! × 3! × 2! × 2!) = 39916800/144 = 277200. Trap 554400 divides by only one of the two pairs — the G's or the I's, but not both. Trap 1108800 = 11!/(3! × 3!) fixes the triples and misses both pairs. Trap 138600 slips in one extra division by 2.