Combinatorial Marble Choosing A bag contains $3$ red marbles, $3$ green ones, $1$ lavender one, $6$ yellows, and $4$ orange marbles.
How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?
I am getting $990$ as an answer but that is incorrect. Please help
 A: 
A bag contains 3 red marbles, 3 green ones, 1 lavender one, 6 yellows, and 4 orange marbles.
How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?

It depends on whether you can tell the difference between marbles of the same color.
Assuming the marbles of the same color are indistinguishable: 
The set is built by selecting either 1 lavender or 1 yellow (2 choices), and then 4 other marbles chosen from the 3 red, 3 green, and 4 orange marbles.  The four others can be selected by:


*
*Choosing a quadruple: ${1 \choose 1} = 1$ choice (all orange)
*Choosing a tripple and a single: ${3 \choose 1}{2 \choose 1}= 6$ choices.
*Choosing a double and two singles: ${3 \choose 1}{2\choose 2} =3$ choices.
*Or choosing two doubles: ${3 \choose 2}=3$ choices

$$2\times(1+6+3+3) = 26\text{ distinct sets}$$

However, if marbles of the same color are distinguishable:
The set is built by selecting either the 1 lavender or 1 of the 6 yellow, and the 4 other marbles are chosen from the remaining 10:
$$\left(1+{6\choose 1}\right)\times{10\choose 4}= 1470 \text{ distinct sets}$$
