Use permutations to solve the following problem. If 5 plants thrive in shade and Asher’s garden has three shaded spots, then how many different arrangements of 3 could he choose out of those 5 plants?(1 point)

Responses

60
60

360
360

10
10

20

1 answer

To find the number of different arrangements of 3 plants that Asher can choose from 5 plants (taking into account the order of the arrangements), we can use the formula for permutations:

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

Where:

  • \( n \) is the total number of items to choose from (in this case, 5 plants),
  • \( r \) is the number of items to arrange (in this case, 3 plants).

Substituting the values into the formula:

\[ P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} \]

Calculating the factorials:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] \[ 2! = 2 \times 1 = 2 \]

Now substituting these values back into the permutation formula:

\[ P(5, 3) = \frac{120}{2} = 60 \]

Therefore, the number of different arrangements of 3 plants out of the 5 is 60.

So the correct response is:

60