Kisha has five different marbles. How many different ways can she place the marbles into two bags so that at least one marble is in each bag?

Assuming the 2 bags are distinguishable.

then :
C(5,1)xC(4,4) + C(5,2)xC(3,3) + C(5,3)xC(2,2) + C(5,4)xC(1,1)
= 5 + 10 + 10 + 5
= 30

notice that this is row 1 5 10 10 5 1 of Pascal's triangle with out the leading and ending 1's.

If the 2 bags are indistinguishable, then there are half of the above, or 15