# The board of directors of a pharmaceutical corporation has 10 members. Three members of the board of directors are physicians. How many different slates consisting of a president, vice president, secretary, and treasurer be made with exactly one physician. Note that there are 3 physicians out of the 10 directors.

The board of directors of a pharmaceutical corporation has 10 members. Three members of the board of directors are physicians. How many different slates consisting of a president, vice president, secretary, and treasurer be made with exactly one physician. Note that there are 3 physicians out of the 10 directors.
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Step 1
You must consider also all the ways to arrange the people into the positions.
Step 2
You have 4 distinct positions to fill, choose 1 of 3 physicians, and 3 of 7 other members (non-physician) to fill the positions.
$4!×{}^{3}{C}_{1}×{}^{7}{C}_{3}=4×3×\left(7×6×5\right)$