How many different permutations can be created when anne, Becky,Carlo, dan and Esi line up to buy movie tickets, if Esi always stand behind becky?
If Esi is always behind Becky, we have only 4 independent choices for each position.
So for the first position, there are 4 choices, for the second, 3 choices left, then 2, then 1.
The number of permutations is the product of the 4 numbers.
Well, it seems like Esi has a thing for following Becky around! In this case, we can consider Esi and Becky as a single unit, so they always stick together. That leaves us with Anne, Carlo, Dan, and the dynamic duo of Esi and Becky.
Now, let's see how many ways we can arrange this lineup. Since there are 4 entities remaining (Anne, Carlo, Dan, and Esi & Becky), we can arrange them in 4! (factorial) ways.
However, within the Esi & Becky unit, there are two possibilities: either Esi can stay behind Becky or Becky can stay behind Esi. So, for each of the 4! arrangements, there are two options for positioning Esi and Becky.
Hence, the total number of different permutations would be 4! * 2 = 48! I guess Esi just can't resist following Becky anywhere.
To solve this problem, we need to fix the position of Esi behind Becky. Since the remaining 4 people can line up in any order, we need to find the number of permutations for those 4 people.
First, let's fix Esi behind Becky. This reduces the problem to permuting the remaining 4 people: Anne, Carlo, Dan, and Becky.
The number of permutations of 4 people can be calculated as 4 factorial (4!).
4! = 4 × 3 × 2 × 1 = 24
Therefore, there are 24 different permutations when Anne, Becky, Carlo, Dan, and Esi line up to buy movie tickets, with Esi always standing behind Becky.
To find the number of different permutations when Anne, Becky, Carlo, Dan, and Esi line up to buy movie tickets with Esi always standing behind Becky, we can follow these steps:
Step 1: Fix Esi and Becky's positions.
Since Esi always stands behind Becky, we can consider them as a single entity for now. Hence, we fix their positions by treating them as one.
Step 2: Arrange the remaining three people (Anne, Carlo, and Dan).
Now we have three people remaining to be arranged: Anne, Carlo, and Dan. Suppose we label them as A, C, and D, respectively.
Step 3: Calculate the number of permutations for the three remaining people.
Since we have three options for the first position (A, C, or D), two remaining options for the second position, and the last person takes the last available position, we multiply these options together: 3 x 2 x 1 = 6.
Step 4: Combine the results from steps 2 and 3.
We take the number of permutations for the three remaining people (6) and multiply it by the number of possible positions for Esi and Becky (2), as they are treated as a single entity. Therefore, the total number of permutations is: 6 x 2 = 12.
Hence, there are 12 different permutations when Anne, Becky, Carlo, Dan, and Esi line up to buy movie tickets, with Esi always standing behind Becky.