In how many ways can all the letters of the word BANANA be arranged so that the two N’s never come together? (Each letter is used exactly once; letters that repeat are identical.)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 40
"Never together" is found as all arrangements minus the arrangements with the two N’s together. Repeated letters are allowed for by dividing by the factorial of each repeat.
BANANA has 6 letters: A three times, N twice, B once
All arrangements \(\displaystyle =\frac{6!}{3!\times2!}=\frac{720}{12}=60\)
N’s together: tie NN into one block; the units are [NN], A, A, A, B, that is 5 units with A three times
Arrangements with the N’s together \(\displaystyle =\frac{5!}{3!}=\frac{120}{6}=20\) (the two N’s inside the block are alike, so they give no new order)
N’s never together \(=60-20=40\)
Why the other options are incorrect
Option (1) 60 — Incorrect. This is all the arrangements of BANANA, including the 20 with the N’s side by side.
Option (2) 20 — Incorrect. This is the number of arrangements in which the two N’s are together.
Option (4) 120 — Incorrect. It is more than the 60 arrangements of BANANA in all, so it is impossible.