In how many ways the letters of word ‘PERSON’ can be arranged in the following way

combinatorics

Question: In how many ways the letters of word 'PERSON' can be arranged in the following way such that No row is empty and no letter is repeated

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I was trying to solve this question, but my answer is getting wrong somewhere.
I know the actual solution, but it was a try to solve it in a different way below:

first select 3 words and arrange them in all rows so that we ensure no row is empty,

$\implies$ Total ways to do so is C($6$,$3$) $\cdot2\cdot2\cdot4\cdot3!$

Now select other words and arrange them without any restriction

$\implies$ Total ways = $5\cdot3\cdot4$

Therefore, Total no. of ways = $115,200$

Where I am going wrong?

Best Answer

Count the ways without restriction, subtract the ways that violate the restriction.

You can only violate it by leaving either row 1 or 2 empty.

Without restriction you have $P(8,6)$ arrangements. There are $2 \cdot 6!$ arrangements that violate the restriction.

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