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I am a novice at discrete mathematics and I have been working on trying to get my combinatorical skills up and i was working on some practice questions for permutation practice and i came across this question that is seemingly simple but i dont know how to do this.

A school election is to be heard for 3 faculty and 2 student seats. The faculty with the highest votes gets three years contract, second highest gets two years and the third highest gets a year. Each student seat is for a year. If there are 9 faculty members and 7 students on the ballot, how many different election results are possible assuming that ties do not occur.

I tried calculating the r-combination and adding and multiplying them to see if i get the answer. However, i have had no success.

The answer is 10,584 and i dont know how to get it.

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Because the three faculty positions are for different lengths of time, it matters who gets which one. There are $9$ possible outcomes for the three-year position; once that’s been filled, there are $8$ possible outcomes for the two-year position; and then there are $7$ possible outcomes for the one-year position. Thus, the faculty members can be chosen in $9\cdot8\cdot7$ different ways.

The two student positions, however, are indistinguishable: all that matters is which two of the $7$ students are elected. There are $\binom72$ possible pairs of students from the $7$ on the ballot, so there are $\binom72$ possible ways to choose the student members.

Putting the pieces together, we have

$$9\cdot8\cdot7\cdot\binom72=9\cdot8\cdot7\cdot21=10,584$$

different outcomes.

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Calculate the product of the following:

  • The number of ways to choose $3$ out of $9$ faculties, which is $\binom93=84$
  • The number of ways to arrange them in different order, which is $3!=6$
  • The number of ways to choose $2$ out of $7$ students, which is $\binom72=21$
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$9$ options for number one faculty , $8$ options for number two, $7$ options for number three. $\frac{7\cdot6}{2}$ options for two students.

$\frac{9\cdot8\cdot 7\cdot 7\cdot 6}{2}=10584$

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