This is some AIME question I believe...

What is the answer?

i think 36

By brute forcing, i got 1278 unordered pairs.

The cards here aren't important. The thing they show is important as the question says for example if I get 3 cards$\{\Delta A1,\square B2,\odot C3\}$ it is counted only once and so I make a line solid and continue with the rest in move. That is I have 3 boxes labeled A B C and they are fixed in position. I can place 3 numbers in the first box then 2 in the B and 1 in A. Then we have for shapes 3 in A 2 shapes in B and 1 last shape should be in the last box(box A). and I get $ (3 * 2*1)*(3 * 2*1)= 36.$

BOBTHEGR8 wrote: There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$. In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values. For example we can chose $\{\Delta A1,\square B2,\odot C3\}$ But we cannot choose $\{\Delta A1,\square B2,\Delta C1\}$ In the actual paper, the example that can be chosen is $\Delta A1,\Delta B2,\odot C3\}$ and not $\Delta A1,\square B2,\odot C3\}$. Also, see https://math.stackexchange.com/questions/3439485/a-problem-involving-a-pack-of-27-distinct-cards.

MarricForsberg wrote: The cards here aren't important. The thing they show is important as the question says for example if I get 3 cards$\{\Delta A1,\square B2,\odot C3\}$ it is counted only once and so I make a line solid and continue with the rest in move. That is I have 3 boxes labeled A B C and they are fixed in position. I can place 3 numbers in the first box then 2 in the B and 1 in A. Then we have for shapes 3 in A 2 shapes in B and 1 last shape should be in the last box(box A). and I get $ (3 * 2*1)*(3 * 2*1)= 36.$ Wrong!!! Festus wrote: By brute forcing, i got 1278 unordered pairs. Correct. Festus wrote: BOBTHEGR8 wrote: There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$. In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values. For example we can chose $\{\Delta A1,\square B2,\odot C3\}$ But we cannot choose $\{\Delta A1,\square B2,\Delta C1\}$ In the actual paper, the example that can be chosen is $\Delta A1,\Delta B2,\odot C3\}$ and not $\Delta A1,\square B2,\odot C3\}$. Also, see https://math.stackexchange.com/questions/3439485/a-problem-involving-a-pack-of-27-distinct-cards. Sorry my fault, I will edit.

Yes this will be 5 different cases observing the repetition pattern then you have to ensure unordered triples then ans is 1278.It is little bit brute forse.

Festus wrote: By brute forcing, i got 1278 unordered pairs. me too bro

Can anyone pls post the solution by brute force