i need to understand this problem

Question

Karmen and her mother are planting flowers in their backyard. They have 2 rose

bushes, 3 tulip plants and 2 sunflowers. In how many different patterns can these

flowers be planted?

Answer 1

To find the number of different patterns in which the flowers can be planted, you can use the concept of permutations. Simply put, a permutation is a word that describes the number of ways things can be ordered or arranged.

The formula for permutations is given by:

P(n)=n!

Where n! (n factorial) is the product of all positive integers up to n.

In this case, Karmen and her mother have 2 rose bushes, 3 tulip plants, and 2 sunflowers. So, the total number of flowers is 2 roses bushes + 3 tulip plants + 2 sunflowers =7 total flowers

This means that the number of different patterns that these flowers can be planted is 7! (7 factorial).

7!=7×6×5×4×3×2×1

7!=5040

This means there are 5040 different patterns in which the flowers (2 rose bushes, 3 tulip plants, and 2 sunflowers) can be planted in Karmen and her mother's backyard.

Answer 2

This is a factorial problem with repeats. You have 7 total types of flower, with 2. roses, 3 Tulips, and 2 Sunflowers- so you have to exclude repeated terms.

To set it up, you have: 7!

2! 3! 2!.

You can expand these both, and then eliminate duplicates as shown. I'll do it in a few steps using strikethroughs

7 x 6 x 5 x 4 x 3 x 2 x1

2 x 1 x 3 x 2 x1 x 2 x 1

Once you get ride of the 3 x 2 x1, you are left with the following:

7x6x5x4

2x1x2x1 2x1x2x1 simplifies to 4, so you can strike through that as shown.

7x6x5x4

4

You are now left with only 7 x 6 x 5. Multiply that and you get 210. So there are 210 ways to plant the flowers.

Another helpful video for these problems is linked here:

https://www.youtube.com/watch?v=RpHEhJWo094

Cheers,

-Eric