Question

samples spaces: your turn
in a window display at a flower shop, there are 3 spots for 1 plant eac
to fill these 3 spots, emily has 6 plants to select from, each a differen
type. selecting from the 6 plants, emily can make how many possible
display arrangements with 1 plant in each spot?
(note: the positions of the unselected plants do not matter)
a. 3
b. 6
c. 15
d. 120
e. 216

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 3 plants out of 6 in 3 spots. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 6 \) (total plants) and \( r = 3 \) (spots to fill).

Step2: Apply the permutation formula

First, calculate \( n! = 6! = 6\times5\times4\times3\times2\times1 = 720 \) and \( (n - r)!=(6 - 3)!=3!=3\times2\times1 = 6 \). Then, \( P(6, 3)=\frac{6!}{(6 - 3)!}=\frac{720}{6}=120 \). Alternatively, we can think of it as for the first spot, we have 6 choices, for the second spot 5 choices (since one plant is used), and for the third spot 4 choices. So the number of arrangements is \( 6\times5\times4 = 120 \).

Answer:

D. 120