In how many ways can five students
Suppose the students are A,B,C,D and E
and the seats are 1,2,3,4,5,6,7 and 8
be seated in row of eight seat if a certain two students
a) insist on setting next to each other
Suppose A and B insist on sitting together.
Case 1: A sits left of B
There are 7 ways to choose their seats: 1&2,2&3,3&4,4&5,5&6,6&7, and 7&8
That's 7 ways:
Case 2: A sits right of B
That's also 7 ways.
That's 14 ways to place A and B together
There are now 6 seats left and 3 students C,D, and E left to seat.
For each of those 14 ways to seat A and B, there are
6 ways to seat C. That's 14*6 ways to seat A,B and C.
There are now 5 seats left and 2 students D and E left to seat.
For each of those 14*6 ways to seat A,B and C, there are
5 ways to seat D. That's 14*6*5 ways to seat A,B,C and D.
There are 4 seats left and 1 student, E, left to seat.
For each of those 14*6*5 ways to seat A,B,C and C, there are
4 ways to seat D. That's 14*6*5*4 ways to seat A,B,C,D and E.
Answer: 14*6*5*4 = 1680 ways.
b) Refuse to sit next to each other
There are 8 ways to seat A, 7 ways to seat B, 6 ways to seat C,
5 ways to seat D and 4 ways to seat E.
That's 8*7*6*5*4 = 6720 ways anybody can sit in any seat.
We subtract 1680 ways A and B sit together.
Answer 6720-1680 = 5040.
Edwin
