# How many ways to arrange n identical object x and k identical object y where 2 of object y cannot be together?

I'm not looking for a solution to a question, but rather an explanation for a solution.

Question 1: How many ways are there to arrange 12 identical apples and five different oranges in a row so that no two oranges will appear side by side?

The solution: C(13, 5) * 5!, but I don't understand why you have to multiply by 5!. In other words, why do you have to make the 5 oranges distinct?

Question 2: How many ways are there to arrange the letters in VISITING with no pair of consecutive Is?

The solution: C(6,3) * 5!, here 5! represents the arrangement of the non-I's. So why, unlike the first question, do you not need to make the I's distinct by multiplying 3!? Why is the answer not C(6, 3) * 5! * 3!?

What's so different about these 2 questions that you have to make the oranges distinct but not the I's?

• Because the $I$s are not distinct, but you are told you have "five different oranges" so they are distinguishable. Putting orange A before orange B is not the same as putting orange A after orange B, because you can tell AB from BA. – TokenToucan Jan 24 '16 at 19:57
• @CuddlyCuttlefish Oh I see! When I read 12 identical apples I just assumed the oranges are identical as well. Thanks! – helpme Jan 24 '16 at 20:00
• I actually read it that way the first time as well, and usually it is the case that the oranges would be indistinguishable. Kind of a trick question :) – TokenToucan Jan 24 '16 at 20:02