Please answer the following question completely and correctl

Please answer the following question, completely and correctly.

In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people, if a) the bride must be in the picture? b) both the bride and groom must be in the picture? c) exactly one of the bride and the groom is in the picture?

Solution

a) The brides can be put in any one of the 6 positions. From the other 9 they can be placed in any of the remaining 5 positions, which give 9*8*7*6*5. So the total number of ways is

6*9*8*7*6*5=90720.

b) The bride and groom can be placed in 6*5= 30 positions. From the other 8 we select 4 for the remaining 4 positions. The 4 can be placed in 8*7*6*5 ways. So the total is

30*8*7*6*5=50400.

c) If the bride is in the picture then the bride has any one of the 6 position, from the remaining 8(groom excluded), we have8*7*6*5*4 possibilities. So with the bride we have 6*8*7*6*5*4=40320. Exactly same reasoning with the groom. So the total with exactly one of them is

40320 +40320= 80640.