Question

How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors if a bag contains three red marbles, four green ones, one lavender, five yellows, and five orange marbles?

Answer 1

`C(6,1) * C(12,4)`
`= 6 * 495`
`= 2970`

Since there are 6 marbles that are either lavender or yellow, there are C(6,1), that is 6 choose 1 = 6, ways to choose the yellow or the lavender marble, for each of those ways, we need to choose 4 more marbles from the remaining 12 (the 12 that are not yellow or lavender); this can be accomplished in C(12,4) ways:
`C(12,4)`
`= (12!)/(((12-4)!)(4!))`

`= (12*11*10*9)/(4*3*2*1)`

`= 495`

So for each of the 6 ways of choosing a single yellow or lavender marble, there are 495 ways of choosing the remaining 4 marbles.

Therefore there are `6 * 495 = 2970` ways of making the selection you described.

Note that I assume here that, for example, two collections with a lavender and 4 orange marbles, would be considered 2 different sets if the 4 orange marbles were not the same 4 marbles (from the original 5 orange marbles). This would be the normal interpretation for sets, but it might not be what you intended.