Question 1193982: A photograph of a diving team is to be taken with the team members in two rows. The team has four girls and two boys. The back row has four places while the front row has the coach in the middle with one diver to the left and one diver to the right. How many ways can the divers be arranged in the photograph if the boys and girls are to alternate in the back row?
The textbook answer is 96.
Can someone explain the solution of this problem? Thanks.
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
B = boy
G = girl
The back row must alternate in gender.
We could have BGBG or GBGB
Either way, all the boys are in the back since there are only 2 boys on the team.
Case BGBG:
There are 2 ways to arrange the boys in this configuration (for slots 1 and 3)
For the girls in the back row, we have 4 ways to fill the second slot and 4-1 = 3 ways to fill the fourth slot. That gives 4*3 = 12 ways to select the girls for the back row.
So far we have 2*12 = 24 ways to form the back row in the configuration of BGBG.
Then for the front row, we have 2 ways to arrange the remaining girls to go on either side of the coach.
In all, there are 2*24 = 48 ways to arrange all divers such that we have the format BGBG in the back row.
Case GBGB:
As you can probably guess, not much is different here compared to the previous case above. We have the same basic values to deal with.
There are 2 ways to arrange the boys, 4*3 = 12 ways to select the girls in the back row, and 2 ways to arrange the remaining girls up front. So we have 2*12*2 = 48 ways to form case GBGB.
In short, we found 48 ways to have the format BGBG in the back row. And we also found 48 ways to have GBGB in the back row.
Therefore, we have 2*48 = 96 different configurations overall.
Side note: order matters when it comes to problems like this when we arrange people in a photograph.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
(1) The back row must contain both boys and 2 of the 4 girls. We need to choose 2 of the 4 girls to be in the back row. The number of ways to do that is C(4,2) = 6.
(2) We need to arrange the 4 people in the back row so that boys and girls alternate.
The first person in the row can be any of the 4
The second person must be one of the 2 of the opposite gender
The third person has to be the other student of the same gender as the first
The fourth person has to be the other student of the same gender as the second
Number of ways to arrange the 4 people in the back row: 4*2*1*1 = 8
(3) The two girls in the front row can be arranged in 2 different ways.
Total number of ways to arrange the students in the photograph: 6*8*2 = 96
ANSWER: 96
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