If each tile contains different combinations of the numbers 0-5,
1. How many tiles contain exactly two 1’s?
If there are exactly two ones then there cannot be a one in the third corner. If this is so, the number in the third corner must be a 0, 2, 3, 4, or 5. This means there are 5 choices for what is in the last corner, therefore five tiles that have exactly two ones.
(5 nCr 1) =5
2. How many tiles contain exactly one 5?
If there is exactly one 5 then there are 5 other numbers that can be in one corner and 5 numbers that can be in the other corner. Since it does not matter if the numbers in the remaining two corners are the same or different, we can multiply the number of options together because for each value placed in one corner you can have those same five options in the other corner.
(5 nCr 1)(5 nCr 1)= 5*5 = 25
3.How many tiles contain at least one 2?
If there is at least one 2, there can be one 2, two 2’s or three 2’s. In the case of one 2, as seen in the second problem, we can multiply the five options for one corner and the five options for the other corner together to arrive at 25 tiles. In the case of two 2’s, there are only five options for the last corner as seen in the first problem. In the case of three 2’s, this can only occur once with one tile all having 2’s. Add these options together and you get 31.
(5 nCr 1) (5 nCr 1) + (5 nCr 1) + (1 nCr 1)
= (5)(5) + 5 + 1
=25 + 5 + 1
=31
4. How many tiles are there altogether in the game?
The number of tiles that would contain 1 of any of the numbers would be 31. This multiplied by 6 would give you 186. However, there are not 186 tiles because those that contain one number also contain other numbers in which you need to subtract the common tiles to arrive at the number of tiles used in the game.
Number of 0’s: 31
Number of 1’s: 31 – (5 + 5) = 31-10 = 21
Number of 2’s: 31 – (10) – (4+4) =31-18 = 13
Number of 3’s: 31 – (10) – (8) – (3+3) = 31-24 = 7
Number of 4’s: 31 – (10) – (8) –(6) – (2+2) = 31-28 = 3
Number of 5’s: 31 – (10) – (8) –(6) – (4) – (1+1) =31-30= 1
When you add all of these values together you get that there are 76 tiles in the game.