Count the number of distinct 8-digit words that may be made by permuting a given multi-set

given that zero is not allowed in either the leftmost place or the rightmost place.

The given multiset is


or alphabet


I started the problem by dividing the S into two subsets

S={0,0,0,1,3,5,8,9} cardinality = 8

F={1,3,5,8,9} cardinality = 5

Now, my professor is a stickler for showing proof when showing work. Absolutely no English words allowed in the proof.

I got Ocp ε SxF {a1,a2,a3,a4,a5,a6,a7,a8|F Ocp a1, F Ocp a8, S Ocp a2…7}


=(5^2)(8^6)/3! .

Does the logic of this proof track?

Also, Ocp = occupies.

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