How many ways are there to assign each of 6 friends to either the chemistry class or the biology class if one of these six, Manoj, refuses to be in a class without any of his friends?
Yes there are many more than 8 possibilities.
I think I will assign the 5 first and then assign the sixth one later.
number of ways that there can be all 5 in one subject is 1*2 = 2 (because there are 2 subjects)
Manoj would have to be with the others so when he is added there are still only 2 ways
------
number of ways that there can be 1 in one subject and 4 in the other in one subject is 5*2 = 10
number of ways that there can be 2 in one subject and 3 in the other in one subject is 5C2*2 = 10*2=20
So that is 30 ways the 5 can be assigned if they are not all in the same class.
with each of these Manoj can be in either class so when I add him the number of possibilities double.
So that is 60 ways where they are not all in the same class.
--
Add the other 2 and I think there are 62 ways that the children can be placed into subjects where Manoj is not left on his own.
------
If you have been given an answer then please let me know what it is. If it is different from mine maybe we can work out why.