A bouquet of roses is to be composed by a florist. Six roses are chosen at random from four red roses and five white roses. In how many ways can the bouquet of roses be composed if there must be more white roses than red roses?
To determine the number of ways in which the bouquet of roses can be composed, we need to consider the following conditions:
1. There must be more white roses than red roses.
2. We need to choose six roses in total.
Let's break down the problem into a few steps:
Step 1: Determine the minimum number of red or white roses in the bouquet.
Since there must be more white roses, we can have a minimum of one red rose and five white roses. We will consider this as the base case and then increment the number of white roses while keeping at least one red rose.
Step 2: Count the ways to choose the roses for each base case.
We start with the base case of one red rose and five white roses.
- Choose 1 red rose out of 4: This can be done in 4C1 ways.
- Choose 5 white roses out of 5: Since we already have 5 white roses, this can be done in only 1 way.
So, for the base case of one red rose and five white roses, there are 4C1 x 1 = 4 ways.
Step 3: Repeat step 2 for different numbers of white roses.
We need to increment the number of white roses while ensuring that there is still at least one red rose in the bouquet.
For two white roses, we have:
- Choose 1 red rose out of 4: This can be done in 4C1 ways.
- Choose 2 white roses out of 5: This can be done in 5C2 ways.
So, for the case of two white roses and one red rose, there are 4C1 x 5C2 = 4 x 10 = 40 ways.
For three white roses, we have:
- Choose 1 red rose out of 4: This can be done in 4C1 ways.
- Choose 3 white roses out of 5: This can be done in 5C3 ways.
So, for the case of three white roses and one red rose, there are 4C1 x 5C3 = 4 x 10 = 40 ways.
For four white roses, we have:
- Choose 1 red rose out of 4: This can be done in 4C1 ways.
- Choose 4 white roses out of 5: This can be done in 5C4 ways.
So, for the case of four white roses and one red rose, there are 4C1 x 5C4 = 4 x 5 = 20 ways.
Step 4: Add up all the possible ways from each base case.
To get the total number of ways in which the bouquet of roses can be composed, we need to add up all the ways from each base case.
Total ways = 4 + 40 + 40 + 20 = 104 ways.
Therefore, there are 104 different ways in which the bouquet of roses can be composed if there must be more white roses than red roses.
the possibilities are, of course,
0R 6W
1R 5W
2R 4W
So, can you determine how many ways for each of those setups?