Questions: The Student Nurses' Association at Seminole State must appoint officers including a president, vice-president, secretary and treasurer. It must also appoint a planning committee with four different members. There are 16 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below. a. How many different ways can the officers be appointed? There are different ways to appoint the officers. b. How many different ways can the committee be appointed? There are different ways to appoint the committee. c. What is the probability of randomly selecting the committee members and getting the four youngest of the qualified candidates? P (getting the four youngest of the qualified candidates) = (Type an integer or a simplified fraction.)

Solution Steps

To solve the given problem, we need to address each part separately:

a. To find the number of different ways to appoint the officers, we need to consider the permutations of 16 candidates taken 4 at a time (since the order matters for positions).

b. To find the number of different ways to appoint the committee, we need to consider the combinations of 16 candidates taken 4 at a time (since the order does not matter for committee members).

c. To find the probability of randomly selecting the committee members and getting the four youngest candidates, we need to calculate the number of favorable outcomes (which is 1, since there is only one way to choose the four youngest) divided by the total number of ways to choose 4 candidates out of 16.

To determine the number of different ways to appoint the officers (president, vice-president, secretary, and treasurer) from 16 candidates, we use the permutation formula:

\[ P(n, r) = \frac{n!}{(n - r)!} \]

where \( n = 16 \) and \( r = 4 \). Thus, the calculation is:

\[ P(16, 4) = \frac{16!}{(16 - 4)!} = 16 \times 15 \times 14 \times 13 = 43680 \]

To find the number of different ways to appoint a planning committee of 4 members from the same 16 candidates, we use the combination formula:

\[ C(n, r) = \frac{n!}{r!(n - r)!} \]

where \( n = 16 \) and \( r = 4 \). Thus, the calculation is:

\[ C(16, 4) = \frac{16!}{4! \times (16 - 4)!} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820 \]

To find the probability of randomly selecting the four youngest candidates for the committee, we calculate:

\[ P(\text{four youngest}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1}{1820} \]

This results in:

\[ P(\text{four youngest}) \approx 0.0005495 \]

Final Answer