How many different arrangements of all the letters of the word PLANET begin with the letter P? (Each letter is used exactly once.)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 120
When a letter is fixed in a place, only the other letters are arranged. \(n\) different letters can be arranged in \(n!\) ways.
PLANET has 6 letters, all different: P, L, A, N, E, T
P is fixed in the first place, so 5 letters (L, A, N, E, T) are left for 5 places
Arrangements \(=5!=5\times4\times3\times2\times1=120\)
Why the other options are incorrect
Option (1) 24 — Incorrect. This is \(4!\), as if only 4 letters were left after P; but 5 letters are left.
Option (2) 720 — Incorrect. This is \(6!\), all the arrangements of PLANET, including those that do not begin with P.
Option (4) 600 — Incorrect. This is \(720-120\), the arrangements that do not begin with P.