"In how many different ways can the letters in the word ‘EMPIRE’ be ordered such that the first letter is a vowel? (HARD)
Answer choices: 60, 120, 180, 240, 360.
GREGMAT solution:
It should be 360, right? Because 3 (it could be any among E, E, I) × 5 × 4 × 3 × 2 × 1 × 2 (since it could be EI or IE) = 720 / 2! = 360? Or I could be wrong? @Leaderboard, your help is much appreciated.
You divide by 2 because the E’s are not distinguishable, and while doing 3 \cdot 5! you considered it to be.
As mentioned above, it has to do with the two E’s and not really anything to do with the I.
Suppose that I is the first vowel, then you would be counting the number of distinct arrangements of (E_1, M, P, R, E_2). Since E_1= E_2 then arrangements like (E_1,E_2,M,P,R) isn’t any different from (E_2, E_1, M, P, R) because they’re both just (E, E, M, P, R).
Perhaps that should motivate why greg divided by 2.
Just to reiterate, this has already been accounted for and so multiplying by 2 again just overcounts everything. The only relevant “scaling” happens to be dividing by 2 for the aforementioned reasons.
Got it, thanks.
