<p>A set of <img alt="$12$" height="12" src="https://latex.artofproblemsolving.com/e/d/f/edf074831eb5bc9e61d6d6e09f525a86e3068f6a.png" width="17" /> tokens — <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" /> red, <img alt="$2$" height="12" src="https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png" width="8" /> white, <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> blue, and <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" /> black — is to be distributed at random to <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" /> game players, <img alt="$4$" height="12" src="https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png" width="9" /> 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 <img alt="$\frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. What is <img alt="$m+n$" height="12" src="https://latex.artofproblemsolving.com/0/2/d/02dbab3601fa6cce490d54263017048f69e1a35d.png" width="48" />?</p>

Question No: 346

A set of $12$ tokens — $3$ red, $2$ white, $1$ blue, and $6$ black — is to be distributed at random to $3$ game players, $4$ 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 $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

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Tags: Combinatorics Probability

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