7 people (A, B, C, D, E, F and G) go to a movie and sit next to each other in 7 adjacent seats in the front row of the theater.How many different arrangements are possible? If A will not sit to the left of F and F will not sit to the left of E. How many different arrangements are possible.
Question
7 people (A, B, C, D, E, F and G) go to a movie and sit next to each other in 7 adjacent seats in the front row of the theater.How many different arrangements are possible? If A will not sit to the left of F and F will not sit to the left of E. How many different arrangements are possible.
Solution
The total number of arrangements for 7 people sitting in 7 seats is 7!, which equals 5040.
However, we have two conditions: A will not sit to the left of F and F will not sit to the left of E.
We can consider A, F, and E as a single group. This group can be arranged in 3! = 6 ways. The remaining 4 people (B, C, D, G) can be arranged in 4! = 24 ways.
So, the total number of arrangements for this group of 7 people, considering A, F, and E as a single group, is 5! * 3! = 720 * 6 = 4320.
But, we have over-counted the arrangements where A is to the left of F or F is to the left of E within the group of A, F, and E.
We need to subtract these arrangements.
If A is to the left of F within the group, we can consider A and F as a single group. This group can be arranged in 2! = 2 ways. The remaining 5 people (B, C, D, E, G) can be arranged in 5! = 120 ways. So, the total number of arrangements for this case is 6! * 2! = 720 * 2 = 1440.
Similarly, if F is to the left of E within the group, we can consider F and E as a single group. This group can be arranged in 2! = 2 ways. The remaining 5 people (A, B, C, D, G) can be arranged in 5! = 120 ways. So, the total number of arrangements for this case is also 1440.
So, the total number of arrangements where A will not sit to the left of F and F will not sit to the left of E is 4320 - 1440 - 1440 = 1440.
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