Risk without Wilds

There are $42$ territory cards in the deck—$14$ that depict an infantry unit, $14$ that depict a cavalry unit, and $14$ that depict an artillery unit. Once you have three cards that either (1) all depict the same kind of unit, or (2) all depict different kinds of units, you can trade them in at the beginning of your next turn in exchange for some bonus units to be placed on the board.

If you are randomly dealt three cards from the $42$, what is the probability that you can trade them in?

In this case, we see that there are $\binom{42}{3}=11480$ possible hands combined of three Risk cards if you remove the wild cards. There are $\binom{14}{3}=364$ possible combinations of three cards within any kind and three different kinds (that is, infantry, cavalry and artillery), so a total of $1092$ different three of a kind sets that can be formed. There are also $14^3 = 2744$ different one of each sets. So there a total of $3836$ sets of three Risk cards without wild cards, which implies that the probability of being able to trade in a randomly dealt set of three cards when the wild cards have been removed is $$p = \frac{3836}{11480} = \frac{137}{410} \approx 33.41463\dots \%$$