Tom asked in Science & MathematicsMathematics · 1 decade ago

# Possible arrangements of Scrabble tiles?

One for the mathematicians...

Assuming a standard game of Scrabble, how many different combinations of tiles are there on a playing board (regardless of whether they spell words), and is it possible that with all the Scrabble games sold that each and every one of those layouts has been seen, or if not will it ever be possible assuming the indefinite survival of the human race and Scrabble?

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• 1 decade ago
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So it doesn't matter if they're valid words or legal Scrabble plays or anything? Well, there are 15×15 = 225 squares on the board. There are 100 tiles.

That means there are 225 choose 100 ways to lay out the tiles, regardless of which goes where. Then you account for the number of ways to rearrange the tiles. There are 100 tiles, and assuming we use the English version, there are:

100! / (12! 9!^2 8! 6!^3 4!^4 3! 2!^9)

ways to arrange them. That gives a total of:

(225 C 100) 100! / (12! 9!^2 8! 6!^3 4!^4 3! 2!^9)

Which is:

69 139 639 564 340 292 641 510 502 298 575 599 642 366 755 460 421 556 758 471 217 502 993 280 859 440 629 040 143 402 231 859 506 703 999 704 847 949 522 719 653 755 277 975 023 059 824 793 702 832 166 755 704 483 348 480 000 000 000 000 000

Or approximately:

69 × 10^180

It is impossible for everyone to have seen them for the following reasons:

(1) Most of these arrangements are invalid. I doubt anyone would ever see a scrabble board with ZQWEEX on it.

(2) There are just too many, unless you seriously want us to allow for "infinite" time (at which point, the probability of anything like this approaches 1).

If there were a trillion universes, each with a trillion galaxies, each with a trillion trillion trillion different planets, each with 100 billion people, and every person saw two new Scrabble boards every second (one in each eye), it would still take:

10^103 years

The age of the universe is only 13 billion years. Compare the two:

10^103 > 1.3 × 10^9

Way bigger.

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