Combinatorics: Questions Involving Internal Orderings
In how many ways can the letters M, O, T, O, and R be arranged so that the two O's are *not* adjacent?
Correct.
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First, find the number of total cases. The number of ways to arrange five letters of which two are identical equals the **number of arrangements for five different letters** divided by the number of **internal arrangements of the two identical letters** (which you don't want to count, since they do not yield distinct arrangements):
> $$\text{Total} = \dfrac{5!}{2!}= \dfrac{5\times4\times3\times2\times1}{2\times1} = 60$$.
Next, find the number of forbidden cases (in which the two O's *are* next to each other). Treat the two O's as a single unit. Now you are arranging four distinct items: "OO", M, T, and R. There are $$4!$$ ways to arrange four distinct items:
>$$\text{Forbidden combinations} = 4! = 24$$
*Note: since the two O's are identical, you need _not_ multiply that by the number of internal arrangements of the O's (2!). There is only one way to arrange "OO".*
Finally, find the number of good cases:
>$$\text{Good combinations} = 60 - 24 = 36$$.
Incorrect.
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You may have gotten this answer if you forgot to consider the fact that there are duplicate O's when calculating the total cases.
Incorrect.
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Incorrect.
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Incorrect.
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This is the number of arrangements where the O's *are* adjacent. *(Read the question carefully!)* You should subtract this from the total cases.
24
36
48
72
96
