My wife and I have a set of dominoes that go up to double 12s. (For those not familiar with this game, it consists of a number of tiles, where each tile's face is divided into two parts. Each part can have one to twelve spots (or pips I think they are called) or can be blank.

While playing yesterday, I got to thinking how many tiles would make a complete set. The box that the tiles are stored in says there are 91 tiles in this set.

Can anyone come up with the reason for this number?
Mark

 

Looks good to me.

Another way to think of it is combinatorially--the number of combinations of 13 symbols (blank, 1, 2, ..., 12) taken two at a time. A symbol for this is 2C13, which is defined to be equal to 13!/(11! * 2!), where ! denotes factorial.

13!/(11! * 2!) = (13 * 12)/2 = 78.

This counts all of the tiles with different symbols on them. To get the double tiles, you need to add 13 to the result above, which gives you 91.
Mark

 

A 12-pip set! Never come across one of them before. I regularly play 5s and 3s and we use 6-pip sets and occasionally 8-pip sets. Will have to look out for one on the markets.

13!/(11! * 2!) = (13 * 12)/2 = 78.

This counts all of the tiles with different symbols on them. To get the double tiles, you need to add 13 to the result above, which gives you 91.

For what it is worth, this is how I would have gone about this.

Dave

 

For what it is worth, this is how I would have gone about this.

I have to confess that my first thoughts were along the lines of the tabular counting method. My guest had given some thought to this problem before, and noticed that 7 * 13 = 91, which made him think of 14C2, the number of ways of combinations of 14 things taken 2 at a time. I couldn't think of any way to justify the 14. 13C2 seemed to make sense, but produced 78, a number that was too small. His insight that we needed to add the 13 double tiles separately gave us the answer we were looking for.

 

I have to confess that my first thoughts were along the lines of the tabular counting method. My guest had given some thought to this problem before, and noticed that 7 * 13 = 91, which made him think of 14C2, the number of ways of combinations of 14 things taken 2 at a time. I couldn't think of any way to justify the 14. 13C2 seemed to make sense, but produced 78, a number that was too small. His insight that we needed to add the 13 double tiles separately gave us the answer we were looking for.

Yes it is simple to be blinded by something simple.

This reminds me of a common pub-quiz question we have here in the UK: the question is between 1 and 100 how many times does the number 7 (it can be any single digit number) appear in total? The amount of times no-one in the pub gets it right is staggering!

Also not sure why zbst deleted their good answer to the the question in the OP

Dave

 

Also not sure why zbst deleted their good answer to the the question in the OP
Dave

Yeah, that's too bad. That answer was perfectly valid.