- #1

stfz

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## Homework Statement

In how many ways can 4 red balls, 3 white balls and 1 black ball be arranged in a line so that the black ball is always surrounded by a red and a white ball?

**Textbook answer: 20**

## Homework Equations

Chapter was on permutations and combinations, so

##n P r = n!/(n-r)!##

##n C r = n!/(r!(n-r)!)##

## The Attempt at a Solution

We have 4 red, 3 white, 1 black.

Black must be surrounded by 1 red and 1 white, so we treat RBW as one unit.

There are two possible combinations of this unit, RBW and WBR, so the result will be multiplied by 2.

There are 6 possible positions to place this 3 ball unit into an 8-ball space, so there are 6*2 = 12 different ways to place this unit.

For the remaining, we regard only order of color, so we use nCr:

For the red balls (only 3 remaining, and we have 5 spaces) there are 5 C 3 = 10 different ways to place them.

For the white balls (only 2 remaining, and we have 2 spaces), there are 2 C 2 = 1 ways to place them.

So the final result is : 12 * 10 * 1 = 120.

I would like a confirmation that this answer is correct (I'm usually not, but maybe I am just this once )