In how many ways the letters of the word "LOCKDOWN" can be arranged

It means N is between two O's not between two O's only.

LOCKDOWN
cases: OCKDWNOL, ONOCKDWNL and so on. Every third number will be like that. Hence division by 3.

I still don't understand why every three permutations will have an 'N' between two 'O's

Could you please elaborate on how to find every 3rd case would have N between?

TIA!

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because you can have ONO, OON, or NOO with spaces between each or not
only ONO will have N in between, so 1/3 of the options

Can you elaborate a bit more on Now in 1/3rd of the cases N will come between O's. this specific part?­

I think this question is tougher than any that I've seen in under the "Elementary Combinatorics" heading of the OGs since about the year 2000. I doubt you'd get a question like this on the test anymore. Alas, I'm a nerd and it's fun, so here goes.

If we ignore the ONO limitation, we have 8!/2!.

How many of those arrangements will have the N before both of the Os? Well, if we ignore the other letters and just look at the N and two Os, the N will be first 1/3rd of the time.
Similarly, the N will be between the Os 1/3rd of the time.
And the N will be after the Os 1/3rd of the time.

We are only allowed to include arrangements where the N is between the Os, so we need 1/3rd of 8!/2!.

8*7*6*5*4 = 56*120 = 56*100 + 56*20 = 5600 + 1120 = 6720

Answer choice E.­

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