Dreidel Coloring

Probability Level 4

A dreidel is a top with 4 faces, with a distinct Hebrew alphabet written on each face.

We are given 13 different colors, and want to color each face and each alphabet on the dreidel such that

each alphabet has a different color;
the color of the alphabet is different from the color of its face.

In how many ways can we color the dreidel?

Notation: $P_k^r$ denotes $k$ -permutation of $r$ . Mathematically, $P_k^r = \dfrac{r!}{(r-k)!}.$

Image Credit: My Jewish Learning.

\dbinom{13}{4} \cdot 12^4

\dbinom{13}{4} \cdot \dbinom{12}{4}

\dbinom{13}{4} \cdot P_4^{12}

P_4^{13} \cdot P_4^{12}

P_4^{13} \cdot 12^4

P_4^{13} \cdot \dbinom{12}{4}

1 solution

Michael Huang
Nov 30, 2016

Note that there are $12 + 1 = 13$ different possibilities. $1$ comes from leaving the portion unpainted, whereas $12$ comes from choosing a color. Because there is an option to leave any indicated feature unpainted, the number of possibilities is not $12$ . This is valid if every indicated feature is painted.

If there are $13$ colors to choose from, the problem is very easy to look at, which involves the same reasoning for setting up the number.

Since all $4$ distinct alphabets have different colors, we have $\dfrac{13!}{(13-4)!} = P_4^{13}$ possibilities. Then, since each of the $4$ faces of the dreidel cannot display the same color as its alphabet, this implies there are $(13 - 1)^4 = 12^4$ ways to color the faces. Therefore, the answer is $\boxed{P_4^{13} \cdot 12^4}$ .

How many Hebrew alphabets are there?

Calvin Lin Staff - 4 years, 6 months ago

Hehe. 4 Hebrew alphabets. I edit the problem, so viewers can understand it.

Too early for Hanukkah, but I posted the problem for fun. Wish I can speak, read or write Hebrew. :)

Michael Huang - 4 years, 6 months ago

The problem statement has been edited. I hope it doesn't change the original meaning?

Christopher Boo - 4 years, 6 months ago

It changes the original meaning a bit. I mean to say that there are 12 choices of colors and possible choice of leaving the feature unpainted. However, I don't think that is the big issue.

Michael Huang - 4 years, 6 months ago

That's the issue we had, where it wasn't obvious that one could leave the side unpainted as an option. Hence it was much better to say that there are 13 colors to choose from, where unpainted is treated as a color.

The problem shouldn't be tricky because "haha you didn't know the side could be unpainted" or "haha you didn't know what a dreidel is". Such problems result in a low attempt and correct rate.

Calvin Lin Staff - 4 years, 6 months ago

@Calvin Lin – Thanks for the response! I understand your point.

Michael Huang - 4 years, 6 months ago

@Calvin Lin – Thanks for the edits. Wasn't happy to attempt it before! (-:

Paul Hindess - 4 years, 5 months ago

Dreidel Coloring

Image Credit: My Jewish Learning.

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