In how many ways can we arrange $4$ white, $3$ black, and $2$ red marbles?

Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

1. Show that there are \({}^{10}C_4\) ways of arranging six similar white and four similar black marbles in line.

Could we label all the marbles, so the whites, for example, become \(W_1, W_2, ... W_6\)?

How many possible ways to order the marbles in line are there now?

Now what happens if we take off the labels? How much duplication is there in our answer?

Find the number of ways of arranging four white, three black, and two red marbles in line, assuming that marbles of the same colour are indistinguishable. [A numerical answer is required.]

Again, could we start by labelling the marbles? How many possibilities are there now?

Now what happens if we take off the labels? How much duplication is there in our answer?

2. A bag contains six white and four black marbles. Find the chance that, if two marbles are drawn together, they are both black.

Approach 1:

Following the idea of part (i), how many ways are there of choosing two marbles from the ten in the bag?

Approach 2

Could we draw these two marbles one after the other, instead of drawing them together?