select the correct answer.
there are 9 applicants for 3 jobs: software engineer, computer programmer, and systems manager. which statement best describes this situation?
a. there are $_{9}p_{3}=504$ ways the positions can be filled because the order in which the applicants are chosen doesnt matter.

b. there are $_{9}c_{3}=84$ ways the positions can be filled because the order in which the applicants are chosen doesnt matter.

c. there are $_{9}p_{3}=504$ ways the positions can be filled because the order in which the applicants are chosen matters.

d. there are $_{9}c_{3}=84$ ways the positions can be filled because the order in which the applicants are chosen matters.

Question

select the correct answer.
there are 9 applicants for 3 jobs: software engineer, computer programmer, and systems manager. which statement best describes this situation?
a. there are $_{9}p_{3}=504$ ways the positions can be filled because the order in which the applicants are chosen doesnt matter.

b. there are $_{9}c_{3}=84$ ways the positions can be filled because the order in which the applicants are chosen doesnt matter.

c. there are $_{9}p_{3}=504$ ways the positions can be filled because the order in which the applicants are chosen matters.

d. there are $_{9}c_{3}=84$ ways the positions can be filled because the order in which the applicants are chosen matters.

Accepted Answer

# Brief Explanations:
1. First, understand permutations and combinations: Permutations ($_nP_r$) are used when the order of selection matters (e.g., different positions with distinct roles), while combinations ($_nC_r$) are used when order does not matter (e.g., selecting a group without distinct roles).
2. In this problem, the jobs (software engineer, computer programmer, systems manager) are distinct positions. So, assigning different applicants to different jobs means the order of selection (which applicant gets which job) matters.
3. The formula for permutations is $_nP_r=\frac{n!}{(n - r)!}$. For $n = 9$ (applicants) and $r = 3$ (jobs), $_9P_3=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}=9\times8\times7 = 504$.
4. Analyze the options:
   - Option A: Incorrect, because permutations (not combinations) are used when order matters, and it incorrectly claims order “doesn’t matter.”
   - Option B: Incorrect, because combinations are for order - not - matter situations, but here order (job assignments) matters.
   - Option C: Correct, because the distinct job roles mean order of selection (which applicant gets which job) matters, so we use permutations $_9P_3 = 504$.
   - Option D: Incorrect, because it uses combinations (for order - not - matter) when order actually matters.

# Answer:
C. There are $_{9}P_{3}=504$ ways the positions can be filled because the order in which the applicants are chosen matters.

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