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I need to find how many ways you can arrange the characters in Permutation with the $N$ appearing before all of the vowels but after the $P$. I understand that you can place the $N$ and $P$ $C(6,2)$ ways, but I'm not sure how to accommodate for the vowels. I could place the $N$, the $P$, and the vowels $(6,3)$ ways if the vowels were consecutive, but they can be in any order with any of the other letters mixed in. Any help/hints would be greatly appreciated.

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First choose seven slots for $N,P$ and all the vowels. How many ways to do that? $P$ has to go in the first of those and $N$ in the second. How many ways to arrange the vowels in the remaining slots that you chose? How many ways to arrange the remaining consonants in the slots you didn't choose? Note that there are two $t$'s

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Group 1: Permute the vowels $AEIOU$ $5! $ ways, then add $PN$ on front

Group 2: Separately permute the remaining consonants $RMTT$, $\frac {4!}{2!}$ ways

Interleave the two groups choosing $4$ from $11$, $\binom {11}{4}$

Multiply accumulated options

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Firstly forget one of T to make things easy as we can handle that later. Now consider 2 groups. The group after N and the group before N. Find the number of permutations of both the groups by shifting N towards left each time (with the first case as all vowels only in the forst group). Now we just need to combine the leftover T. For that multiply by 11 and divide by 2!

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