# 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?

Feb 9, 2015

$C \left(6 , 1\right) \cdot C \left(12 , 4\right)$
$= 6 \cdot 495$
$= 2970$

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:
$C \left(12 , 4\right)$
= (12!)/(((12-4)!)(4!))

$= \frac{12 \cdot 11 \cdot 10 \cdot 9}{4 \cdot 3 \cdot 2 \cdot 1}$

$= 495$

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 $6 \cdot 495 = 2970$ 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.