In how many ways can all the letters of the word LETTER be arranged so that the two T’s never come together? (Each letter is used exactly once; letters that repeat are identical.)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 120
"Never together" is found as all arrangements minus the arrangements with the two T’s together. Repeated letters are allowed for by dividing by the factorial of each repeat.
LETTER has 6 letters: T twice, E twice, L and R once each
All arrangements \(\displaystyle =\frac{6!}{2!\times2!}=\frac{720}{4}=180\)
T’s together: tie TT into one block; the units are [TT], L, E, E, R, that is 5 units with E twice
Arrangements with the T’s together \(\displaystyle =\frac{5!}{2!}=\frac{120}{2}=60\)
T’s never together \(=180-60=120\)
Why the other options are incorrect
Option (1) 180 — Incorrect. This is all the arrangements of LETTER, including the 60 with the T’s side by side.
Option (2) 60 — Incorrect. This is the number of arrangements in which the two T’s are together.
Option (3) 90 — Incorrect. This is half of all the arrangements; the T’s are together in 60 of the 180, not in half of them.