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Exercise 6.3 · Q16

Q.The board of directors of a pharmaceutical company has 10 members. An upcoming stockholder's meeting is scheduled to approve a new president, vice president, secretary and a treasurer. How many different ways the four can be appointed.

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The four offices are distinct, so this is an ordered selection (permutation) of 4 members out of 10.

[!FORMULA] nPr=n!(n−r)!^{n}P_{r}=\dfrac{n!}{(n-r)!}, where nn = total members available, rr = number of distinct positions to be filled (order matters since each office is different).

  1. Total members n=10n = 10; number of offices to fill r=4r = 4 (President, Vice-President, Secretary, Treasurer — all different posts, so order of selection matters).
  2. Number of ways =10P4=10!(10−4)!=10!6!= {}^{10}P_{4} = \dfrac{10!}{(10-4)!} = \dfrac{10!}{6!}.
  3. 10!6!=10×9×8×7\dfrac{10!}{6!} = 10\times9\times8\times7.
  4. 10×9=90, 90×8=720, 720×7=504010\times9=90,\ 90\times8=720,\ 720\times7=5040. …
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