PSA_LEVEL-UP_QNR4) There are 5 friends: A,B,C,D and E. They wanted to take a group photograph of all of them sitting in a single row.
a) How many distinctly different photographs can be clicked?
b) In how many of these photographs would A be sitting in the middle?
c) In how many of these photographs would A and B be sitting next to each other?
d) In how many of these photographs would A be before B?
e) In how many of these photographs would A,B and C would be sitting next to each other?
a) 5 friends can arrange themselves in photograph in 5! ways = 5 *4 *3 *2 *1 =120
b) Since, A will be sitting in the middle, let us fix A’s place:
_ _ A _ _
Now, rest of the four places can be occupied by rest for friends (B, C, D & E) for 4!.
c) Since, A and B will be sitting next to each other, let us group them together and treat as one unit. Other three friends (C, D, & E) are the other 3 units. Total units=4.
4 units can arrange themselves in 4! ways. Also, A and B can interchange their places- no of ways= 2!
Total number of ways 4! * 2! = 48.
d) Every person has equally likely chances of occupying a certain place. For example, in all the distinct photographs, A will occupy the first place as many times as B will. Thus, we can ascertain that in half of the photographs A will be before B and vice -versa. Total photograph 1/2 * 120 =60.
Task: To sit A before B.
To sit A and B, first task would be to chose two seats out of 5. This can be done in 5C2 = 10 ways.
1 2 3 4 5
_ _ _ _ _
Let the seat chosen are (2,5), we can sit A and B in those seats in ONLY one way. A has to sit in no. ‘2’ and B has to sit in no. ‘5’.
Now that 2 of the seats are occupied, rest 3 would be filled in by rest of the members in 3! ways or 6.
Total number of ways = 10 * 6 = 60.
e) Since A,B & C will be sitting next to each other, let us group them together and treat as one unit. Other two friends (D, & E) are the other 2 units. Total units=3.
3 units can arrange themselves in 3! ways. Also, A,B, C can arrange among themselves- no of ways= 3!
Total number of ways 3! * 3! = 36.