Question

4.
in a window display at a flower shop, there are 3 spots
for 1 plant each. to fill these 3 spots, emily has
6 plants to select from, each of a different 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 since the order of plants in the spots matters (different arrangements are distinct). We need to find the number of permutations of 6 plants taken 3 at a time.

Step2: Recall the permutation formula

The formula for permutations of \( n \) objects taken \( r \) at a time is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n!=n\times(n - 1)\times\cdots\times1 \). Here, \( n = 6 \) (total plants) and \( r = 3 \) (spots to fill).

Step3: Calculate the permutation

First, calculate \( n - r=6 - 3 = 3 \). Then, \( P(6, 3)=\frac{6!}{3!}=\frac{6\times5\times4\times3!}{3!} \). The \( 3! \) terms cancel out, leaving \( 6\times5\times4 = 120 \).

Answer:

D. 120