Q.In how many ways can the letters of the word 'MOBILE' be arranged?

MOBILE has 6 letters, all distinct, so every arrangement uses all 6 in a row — an ordered arrangement of all objects.

First check the letters of MOBILE: M, O, B, I, L, E — 6 letters, and all 6 are different (no letter repeats). Arranging all 6 distinct letters in a row is $^{6}P_{6} = 6!$: …

Not first checking whether any letter repeats before applying $n!$ — this formula only applies directly when every letter is distinct (a word with a repeated letter, which is out of scope for this chapter's problems, wo …