Rachel has 3 red, 5 blue, and one yellow towel. In how many ways can the towels be aranged if:
a) One blue towel must be at each end of the row?
b) All the blue towels must be side by side?
c) The towel's of the same colour must be side by side?
a) One blue towel must be at each end of the row:
To solve this, we fix one blue towel at each end of the row. So, there are 2 spots remaining between the blue towels where the other towels can be placed.
Let's consider the red, blue, and yellow towels separately:
1. Blue towels: We have 3 blue towels, and we have already placed 1 at each end. So, we have 1 remaining blue towel to place in the 2 spots between the blue towels.
2. Red towels: We have 3 red towels remaining to be placed. Since the remaining 3 spots (2 between the blue towels and 1 after the last blue towel) are indistinguishable, we can arrange the red towels in these spots in 3! (3 factorial) ways.
3. Yellow towel: We have 1 yellow towel remaining, and we can place it in the remaining spot.
To find the total number of arrangements, we multiply the number of ways to arrange each group:
Total number of arrangements = Number of ways to arrange blue towels x Number of ways to arrange red towels x Number of ways to arrange yellow towel.
Total number of arrangements = 1 x 3! x 1 = 1 x 6 x 1 = 6.
Therefore, there are 6 ways to arrange the towels if one blue towel must be at each end of the row.
b) All the blue towels must be side by side:
To solve this, we can treat the group of blue towels as a single unit and count the number of arrangements of this unit along with the red and yellow towels.
So, we have 2 units: the group of blue towels (as one unit) and the remaining red and yellow towels.
Let's consider the red and yellow towels separately:
1. Red towels: We have 3 red towels remaining to be placed, and these can be arranged among themselves in 3! (3 factorial) ways.
2. Yellow towel: We have 1 yellow towel remaining, and it can be placed in the remaining spot.
Now, we treat the group of blue towels as a single unit. Since there are 5 towels in the group and they must be side by side, we just consider the arrangements within this group.
Total number of arrangements within the blue towels group = 5! (5 factorial).
To find the total number of arrangements, we multiply the number of ways to arrange each group:
Total number of arrangements = Number of ways to arrange blue towels group x Number of ways to arrange red towels x Number of ways to arrange yellow towel.
Total number of arrangements = 5! x 3! x 1 = 120 x 6 x 1 = 720.
Therefore, there are 720 ways to arrange the towels if all the blue towels must be side by side.
c) The towels of the same color must be side by side:
To solve this, we can treat each group of towels of the same color as a single unit and count the number of arrangements of these units along with the other towels.
So, we have 3 units: the group of red towels, the group of blue towels, and the yellow towel.
Let's consider each unit separately:
1. Red towels: We have 3 red towels, and they must be side by side. So, we consider this group of red towels as a single unit and have 1 way to arrange them.
2. Blue towels: We have 5 blue towels, and they must be side by side. So, we consider this group of blue towels as a single unit and have 1 way to arrange them.
3. Yellow towel: We have 1 yellow towel, and it can be placed anywhere among the other towels. So, we have 9 possible spots to place the yellow towel.
To find the total number of arrangements, we multiply the number of ways to arrange each group:
Total number of arrangements = Number of ways to arrange red towels group x Number of ways to arrange blue towels group x Number of ways to arrange yellow towel.
Total number of arrangements = 1 x 1 x 9 = 9.
Therefore, there are 9 ways to arrange the towels if the towels of the same color must be side by side.