A committee is to be chosen from among $8$ scientists, 6 laypersons and $13$ clerics. If the committee is to have $2$ members from $2$ different backgrounds, how many committees are there?
My question here is when considering the committees should I consider my $8$ scientists as $S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8$, my $6$ laypersons as $L_1,L_2,L_3,L_4,L_5,L_6$ and my 13 clerics as $C_1,C_2,C_3,C_4,C_5,C_6,C_7,C_8,C_9,C_{10},C_{11},C_{12},C_{13}$
So then I could have a committee of scientists and laypersons, resulting in $8\times6 = 48,$ a committee of scientists and clerics, resulting in $8\times13 = 104$, and a committee of laypersons and clerics, resulting in $6\times 13=78$
Therefore the total number of committees is $48+104+78$
Do I need to consider the order of the committee? did my workings make sense?
Given that we are dealing with people rather than, say, colored marbles, I think it is safe to assume that the individual people are to be distinguished from each other, and in that case your answer is correct.
If, however, one scientist is to be seen as any other scientist, and same for the other positions, then there are just three committees possible ... which isn't very interesting ... and so another reason to believe that the question wants you to differentiate between each individual person.
Also, order does not appear to matter ... otherwise the question would have said something like: the committee consists of a chairman and a side-kick ... or something like that!