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can someone explain to me this permutations problem from my desicrete math textbook?

Q:

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.

A:

My answer was (3x7x6x5), but the textbook has this answer 4(3x7x6x5). How come it is multiplying it by 4?

Edit:

I have figured it out, it is 4 because there are 4 positions for the physician and I was too quick in reading and assumed it only meant president position .

Belphegor
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2 Answers2

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Hint: What role does the physician take on?

Semiclassical
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  • I just figured it out too >.<.. I wasn't reading carefully and I thought it meant only the president spot, but it's the netire slate.. facepalm. – Belphegor Jul 20 '14 at 22:37
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You must consider also all the ways to arrange the people into the positions.

You have 4 distinct positions to fill, choose 1 of 3 physicians, and 3 of 7 other members (non-physician) to fill the positions. $$4!\times {^3C_1} \times {^7C_3} = 4\times 3\times (7\times 6\times 5)$$

Graham Kemp
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