Math / Discrete

QuestionUse the fundamental counting principle. Gina must write evaluation reports on 2 hospitals and 4 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 7,560 ways for her to complete the assignment. (Simplify your answer. Type a whole number.)

Studdy Solution

STEP 1

1. Gina needs to choose 2 hospitals out of 9.
2. Gina needs to choose 4 clinics out of 10.
3. The order in which she chooses the hospitals or clinics does not matter.
4. The fundamental counting principle will be used to find the total number of ways.

STEP 2

1. Calculate the number of ways to choose 2 hospitals out of 9.
2. Calculate the number of ways to choose 4 clinics out of 10.
3. Use the fundamental counting principle to find the total number of ways.

STEP 3

To find the number of ways to choose 2 hospitals out of 9, use the combination formula:
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
For choosing 2 hospitals out of 9:
$\binom{9}{2} = \frac{9!}{2!(9-2)!}$

STEP 4

Calculate $\binom{9}{2}$ :
$\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$

STEP 5

To find the number of ways to choose 4 clinics out of 10, use the combination formula:
$\binom{10}{4} = \frac{10!}{4!(10-4)!}$

STEP 6

Calculate $\binom{10}{4}$ :
$\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$

STEP 7

Use the fundamental counting principle to find the total number of ways to complete the assignment by multiplying the number of ways to choose hospitals and clinics:
$36 \times 210 = 7,560$
The total number of ways for Gina to complete her assignment is:
$\boxed{7,560}$

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Studdy Solution

STEP 1

1. Gina needs to choose 2 hospitals out of 9.
2. Gina needs to choose 4 clinics out of 10.
3. The order in which she chooses the hospitals or clinics does not matter.
4. The fundamental counting principle will be used to find the total number of ways.

STEP 2

1. Calculate the number of ways to choose 2 hospitals out of 9.
2. Calculate the number of ways to choose 4 clinics out of 10.
3. Use the fundamental counting principle to find the total number of ways.

STEP 3

To find the number of ways to choose 2 hospitals out of 9, use the combination formula:
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
For choosing 2 hospitals out of 9:
$\binom{9}{2} = \frac{9!}{2!(9-2)!}$

STEP 4

Calculate $\binom{9}{2}$ :
$\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$

STEP 5

To find the number of ways to choose 4 clinics out of 10, use the combination formula:
$\binom{10}{4} = \frac{10!}{4!(10-4)!}$

STEP 6

Calculate $\binom{10}{4}$ :
$\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$

STEP 7

Use the fundamental counting principle to find the total number of ways to complete the assignment by multiplying the number of ways to choose hospitals and clinics:
$36 \times 210 = 7,560$
The total number of ways for Gina to complete her assignment is:
$\boxed{7,560}$

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Studdy Solution

STEP 1

1. Gina needs to choose 2 hospitals out of 9.2. Gina needs to choose 4 clinics out of 10.3. The order in which she chooses the hospitals or clinics does not matter.4. The fundamental counting principle will be used to find the total number of ways.

STEP 2

1. Calculate the number of ways to choose 2 hospitals out of 9.2. Calculate the number of ways to choose 4 clinics out of 10.3. Use the fundamental counting principle to find the total number of ways.

STEP 3

To find the number of ways to choose 2 hospitals out of 9, use the combination formula:(nr)=n!r!(n−r)! \binom{n}{r} = \frac{n!}{r!(n-r)!} (rn​)=r!(n−r)!n!​For choosing 2 hospitals out of 9:(92)=9!2!(9−2)! \binom{9}{2} = \frac{9!}{2!(9-2)!} (29​)=2!(9−2)!9!​

STEP 4

Calculate (92) \binom{9}{2} (29​):(92)=9×82×1=36 \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 (29​)=2×19×8​=36

STEP 5

To find the number of ways to choose 4 clinics out of 10, use the combination formula:(104)=10!4!(10−4)! \binom{10}{4} = \frac{10!}{4!(10-4)!} (410​)=4!(10−4)!10!​

STEP 6

Calculate (104) \binom{10}{4} (410​):(104)=10×9×8×74×3×2×1=210 \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 (410​)=4×3×2×110×9×8×7​=210

STEP 7

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1. Gina needs to choose 2 hospitals out of 9.
2. Gina needs to choose 4 clinics out of 10.
3. The order in which she chooses the hospitals or clinics does not matter.
4. The fundamental counting principle will be used to find the total number of ways.

1. Calculate the number of ways to choose 2 hospitals out of 9.
2. Calculate the number of ways to choose 4 clinics out of 10.
3. Use the fundamental counting principle to find the total number of ways.

To find the number of ways to choose 2 hospitals out of 9, use the combination formula:
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
For choosing 2 hospitals out of 9:
$\binom{9}{2} = \frac{9!}{2!(9-2)!}$

Calculate $\binom{9}{2}$ :
$\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$

To find the number of ways to choose 4 clinics out of 10, use the combination formula:
$\binom{10}{4} = \frac{10!}{4!(10-4)!}$

Calculate $\binom{10}{4}$ :
$\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$