Answer:

There are 840 sets of five marbles that include either the lavender one or exactly one yellow one but not both colors.

Step-by-step explanation:

We need to treat the case where we get the lavender marble, then the case where we get exactly one yellow marble, because according to the question, these cases are mutually exclusive.

Given

Red Marbles = 3

Green = 3

Lavender = 1

Yellow = 3

Oranges = 4

TOTAL = 14

Now

For when we get the lavender marble alone, we have

C(1, 1) = 1!/(0! 1!) = 1/1 = 1 way to do this.

The remaining four marbles are neither lavender nor yellow.

There are 14 - 4 = 10 such marbles, so there are

C(10, 4) = 10!/(6! 4!) = 3628800/17280 = 210 ways to select the last four marbles.

In total, there are

C(1, 1) × C(10, 4) = 1 × 210 = 210 ways to select five marbles where one of them is the lavender color and none of them is yellow color.

Again, we need to get exactly one yellow marble.

There are C(3, 1) = 3!/(2! 1!) = 6/2 = 3 ways to do this.

The other four marbles are neither yellow nor lavender, there are

14 - 4 = 10 such marbles, so we have

C(10, 4) = 210 ways to select the last four marbles.

In total, we have

C(3, 1) × C(10, 4) = 3 × 210 = 630 ways to get exactly one yellow marble.

Finally, using the principle of addition to find the total number of ways to choose a set of five marbles that include either the lavender one or exactly one yellow

one, but not both. We have 210 + 630 = 840 ways to do

this.