Of the nine (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal and Ida), five of them stand in a row for a picture, how many ways can this be done if Ed and Gail are standing next to each other in the picture?
Take out Ed and Gail (7 people left). Choose 5 at a time to take pictures without the couple. There are
N1=7!/(7-2)! possible pictures.
Now treat the couple as a single person, and get 3 other people for pictures (N2).
Don't forget to ask Gail and Ed to change places (2*N2)
Add up the two to get the total=N1+2N2
To determine the number of ways five people can stand in a row for a picture, given that Ed and Gail are standing next to each other, you can use the concept of permutations.
First, let's assume that Ed and Gail are treated as a single entity. We can call this entity "EG" for simplicity. So, now we have eight entities (Ann, Ben, Cal, Dot, EG, Fran, Hal, and Ida) to arrange in a row.
To find the number of ways to arrange these entities, we need to consider two scenarios:
Scenario 1: Ed and Gail are arranged as "EG."
In this case, we treat "EG" as one entity and consider six entities remaining (Ann, Ben, Cal, Dot, Fran, Hal, and Ida). We need to find the number of ways to arrange these six entities.
The number of ways to arrange the six entities is equal to 6! (six factorial) since there are six positions to fill. Mathematically, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720.
Scenario 2: Ed and Gail are arranged as "GE."
Similarly, treating "GE" as one entity, we have the same six entities remaining (Ann, Ben, Cal, Dot, Fran, Hal, and Ida). The number of ways to arrange these six entities is also 6!.
Because Scenario 1 and Scenario 2 are mutually exclusive events (either "EG" or "GE"), we need to sum up the number of ways from each scenario.
Total number of ways = Number of ways in Scenario 1 + Number of ways in Scenario 2
= 6! + 6!
Simplifying this expression, we get:
Total number of ways = 2 * 6!
= 2 * 720
= 1440.
Therefore, there are 1440 ways to arrange the entities in the row for a picture, given that Ed and Gail are standing next to each other.