Stacking Bowls

Probability Level 4

I have a set of six plastic bowls I use to serve food to my three-year-old twins:

They are mostly white, with different colored trim: Two are blue, two are pink, and two are green.

How many ways can these six bowls be stacked so that no two bowls of the same color are touching each other? (One such way is pictured above.)

Details/assumptions:

Consider the bowls of the same color to be identical. So switching the two green bowls, for example, would not create a new arrangement.

The answer is 30.

1 solution

Xuming Liang
Feb 2, 2014

This solution is not elegant but since I've waited for a few days and no solutions are posted, I will present mine:

We will represent an arrangement as symbols going from left to right, being the same as going from bottom to top for the bowls

We will count the number of "arrangement structures" we could have. To see what I mean, we let $X,O,L$ be indicators of the same color. Then we fix $X$ as our beginning "color", finally we will casework with the number of colors between the two $X$ 's.

When we have $1$ color, there's only one when to avoid adjacent colors: $XOXLOL$

For $2$ colors we have two: $XOLXOL$ , $XOLXLO$

$3$ colors we have one: $XOLOXL$

$4$ colors we have one: $XOLOLX$

This gives us $5$ structures in total. Since there's $3*2=6$ ways to assign colors to these indicators, our answer would be $5*6=\boxed{30}$

There's an easier and more straight forward solution using PIE, but I like these "structures" because they are the real reasons that separate each arrangement from being the same.

Xuming Liang - 7 years, 4 months ago

That's what I did :)

Sam Thompson - 7 years, 3 months ago

Stacking Bowls

The answer is 30.

1 solution

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