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A small college with 12001200 total students has a student government of 4040 members. From its members, the student government will elect a president, vice president, secretary, and treasurer. No single member can hold more than 11 of these 44 positions. The permutation formula can be used to find the number of unique ways the student government can arrange its members into these positions. What are the appropriate values of nn and rr?

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Q. A small college with 12001200 total students has a student government of 4040 members. From its members, the student government will elect a president, vice president, secretary, and treasurer. No single member can hold more than 11 of these 44 positions. The permutation formula can be used to find the number of unique ways the student government can arrange its members into these positions. What are the appropriate values of nn and rr?
  1. Identify total number of members: Identify the total number of student government members nn. n=40n = 40.
  2. Determine positions to be filled: Determine the number of positions to be filled rr. r=4r = 4 (president, vice president, secretary, and treasurer).
  3. Use permutation formula: Use the permutation formula P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n - r)!} to calculate the number of unique arrangements.
  4. Plug in values: Plug in the values of nn and rr into the permutation formula. P(40,4)=40!(40−4)!P(40, 4) = \frac{40!}{(40 - 4)!}.
  5. Calculate factorials: Calculate the factorial of nn and (n−r)(n - r). 40!=40×39×38×…×140! = 40 \times 39 \times 38 \times \ldots \times 1 and 36!=36×35×34×…×136! = 36 \times 35 \times 34 \times \ldots \times 1.
  6. Simplify permutation formula: Simplify the permutation formula by canceling out the common terms in the numerator and the denominator. P(40,4)=40×39×38×371P(40, 4) = \frac{40 \times 39 \times 38 \times 37}{1}.
  7. Perform multiplication: Perform the multiplication to find the number of unique arrangements. P(40,4)=40×39×38×37P(40, 4) = 40 \times 39 \times 38 \times 37.

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