Arrangement of all the letters of a word In how many ways can all the letters of the word ‘PERFORMED’ be placed in the $3 \times 3$ grid of squares, such that each square contains exactly one letter and there is at least one vowel in each row and in each column?
 A: First select the places which the vowels are going to occupy. There are only three vowels so there are $3!=6$ ways to choose these (we need to assign to each column the row of that column that is going to have the vowel, these three numbers must be distinct, so $3!$)
Once we do that there are $3$ ways to decide how to place the vowels inside the selected places (Since this is equivalent to selecting the position for the letter O, and we have three options).
This tells us there are $3\cdot 6=18$ ways to place the vowels.
Once the vowels have been placed we have to place the consonants, however we can do this however we want. There are $6$ places remaining and $6$ consonants, so naively we could say there are $6!$ ways to place the consonants. However in reality we have one letter $P$, two letters $R$ one letter $F$ one letter $M$ and one letter $D$.
This means if we consider the $6!$ arrangements we are counting each arrangement twice, since switch the places of the letter $R$ with the other letter $R$ gives us the same arrangement. Hence there are $\frac{6!}{2}=360$ ways to place the consonants (See multinomial coefficient for more information on this).
All in all there $360\cdot18=6480$ ways to place the letters.
A: Hint:
Since there are $3$ vowels and you are dealing with a $3\times3$ grid at least one vowel in each row and column can only be realized if there is exactly one vowel in each row and column. 


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*Find in how many ways the vowels can be placed.

*Find in how many ways the consonants can be placed.


By 1) take into account that not all vowels are distinct.
By 2) take into account that not all consonants are distinct.
