You have three coins in your pocket, each of which can be a penny, nickel, dime or quarter with equal probability. You might have three different coins, three of the same coin or two coins that are the same and one that is different.
Each of these coins can buy you a string whose length in centimeters equals the value of the coin in cents, i.e., the penny buys 1 cm of string, the nickel buys 5 cm of string, etc. After purchasing your three lengths of string, what is the probability that they can be the side lengths of a triangle?
There are $3$ coins and $4$ possibilities for each coin. This yields $4^3 = 64$ total possibilities.
However, there are only $\dfrac{\left(\left(4-1\right) + 3\right)!}{\left(4-1\right)!3!} = \dfrac{6!}{3!3!} = 20$ unique possibilites.
Here is a breakdown.
$4$ of the unique possibilities are $PPP$, $NNN$, $DDD$, and $QQQ$. All are triangles, and occur exactly once each.
$4$ of the unique possibilities are $PND$, $PNQ$, $PDQ$, and $NDQ$. None are real triangles, and occur exactly 6 times each.
The remaining $12$ unique possibilities occur $3$ times each.
$PPN$, $PPD$, $PPQ$, $NND$, $NNQ$, and $DDQ$ are not real triangles.
$PNN$, $PDD$, $PQQ$, $NDD$, $NQQ$, and $DQQ$ are isoceles triangles.