Question

Use the fundamental counting principle.

Gina must write evaluation reports on 3 hospitals and 2 health clinics as part of her degree program in community health services. If there are 9 hospitals and 10 clinics in her​ vicinity, in how many ways can she complete her​ assignment?
There are __ ways for her to complete the assignments

Answer 1

Gina can complete her assignment in 6 ways using the fundamental counting principle.

Step-by-step explanation:

To solve this problem using the fundamental counting principle, we multiply the number of options for each decision. In this case, Gina has 3 options for hospitals and 2 options for clinics. Therefore, the total number of ways she can complete her assignment is 3 * 2 = 6 ways.

Answer 2

Gina can complete her assignment in 3780 ways by choosing 3 out of 9 hospitals and 2 out of 10 clinics, which are calculated using the combination formula for each selection and then multiplied together.

Step-by-step explanation:

To determine the number of ways Gina can complete her assignment using the fundamental counting principle, we need to calculate the combinations for hospitals and health clinics separately and then multiply them together.

For the hospitals, she needs to choose 3 out of 9. This is a combination problem, which is calculated using the formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose. Thus, the number of ways to choose the hospitals is: C(9, 3) = 9! / (3!(9-3)!) = 84 ways.

For the health clinics, she needs to choose 2 out of 10. Similarly, we calculate the number of ways to choose the clinics: C(10, 2) = 10! / (2!(10-2)!) = 45 ways.

To find the total number of ways she can complete her assignment, we multiply the combinations of hospitals with the combinations of clinics: 84 ways × 45 ways = 3780 ways.

Therefore, there are 3780 ways for her to complete the assignments.