Probability of getting 3 cards in the same suit from a deck

Three cards are selected from a standard deck of $52$ cards. Disregarding the order in which they are drawn, the possible outcomes are $\binom{52}{3}$. Out of these, how many include all cards of the same suit (say hearts)? There are $\binom{13}{3}$ ways in which you can get all 13 heart cards.

Since there are 4 suits, there are $4\binom{13}{3}$ ways in which all cards drawn are of the same suit. Thus the probability is:

$$\frac{4\binom{13}{3}}{\binom{52}{3}}\approx 5.18\%$$

Here is another solution that might be easier to compute/reason with.

Your first card can be anything. So you have 52 choices out of 52 cards (because no matter what card you draw you can get a full hand of the same suite).

Your second card, has to be the same suit as your first card, so probability of that is $\frac{12}{51}$ because there are 13 of each suite and you have to subtract 1 for the one card you have drawn.

Your third card has to be the same suite as the first and the second, notice there are only 11 cards left of that suite, so selecting that specific card will be $\frac{11}{50}$

Giving a total probability of:

$$\frac{52}{52} \times \frac{12}{51} \times \frac{11}{50}$$