Determine the number of permutations of $ \ \{1,2,3,4,5,6,7,8,9,10\} \ $ that have exactly 3 numbers in their natural position Determine the number of permutations of $ \  \{1,2,3,4,5,6,7,8,9,10\} \  $  that have exactly  3 numbers in their natural position. $$ $$ Is it $ \ \ \begin{pmatrix} 10 \\ 3 \end{pmatrix} \times 7 ! \  $ ?

No, because $7!$ allows for extra elements to be in their natural position.  After the $3$ fixed position items are selected, you need a derangement of the remaining $7$ items.
