2024 AMC 12A Problems/Problem 16

Problem

A set of tokens ---- red, white, blue, and black ---- is to be distributed at random to game players, tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as , where and are relatively prime positive integers. What is ?

Solution 1 (12fact bash)

We have total possible arrangements of distinct tokens. If we imagine the first tokens of our arrangement go to the first player, the next go to the second, and the final go to the third, then we can view this problem as counting the number of valid arrangements.

Firstly, the tokens are not all distinct, so we multiply by , , , and to account for the fact that the red, white, blue, and black tokens, respectively can switch around from where they are.

Letting denote red, denote white, denote blue, and denote black, then our arrangement must be something like . The three players are arbitrary, so we multiply by ; then, the player who gets the reds has possible arrangements, the player who gets the whites has possibilities, and the player who gets the blacks has possibilities. Our total on top is thus , and the denominator is . Firstly, we have the in the numerator cancel out part of the denominator; we thus have the following:

Our answer is

~Technodoggo

Solution 2

Assume all of them are distinct even though some have the same color,

Total possibility = (choosing 4 random token for each person)

Next, assume that all the token are already in 3 different groups (Note: 3! Ways to do so since 3 people)

We then distribute the 6 distinct black token into these 3 different groups (So 1,2,3 token for each group)

There are a total of ways in doing so

Thus the answer is

So the answer is

~lptoggled

Solution 3

We first assume there are designated red, white, and blue token players that will receive all of their respective one.

Consider each non-black token: The probability of the red player getting the first red token is , because there are each player has 4 empty token "slots" for a total of 18. It follows that the probability of the player receiving all 3 red tokens is , the white token player is , and the blue token player is .

The combined probability is .

Finally, we multiply the probability by to remove our initial assumption to get .

The requested sum is .

~SilverRush

See also

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