Why is a royal flush rarer than a full house in poker?

In poker, the rank of a hand is determined by its mathematical probability; the more unusual or rare a combination of cards is, the higher it is ranked.123

A royal flush is significantly rarer than a full house because there are far fewer ways to form the specific combination required for a royal flush compared to a full house. Specifically, there are only 4 possible ways to be dealt a royal flush from a 52-card deck, resulting in odds of approximately 1 in 649,740. In contrast, there are 3,744 possible ways to form a full house, making the odds of being dealt one approximately 1 in 694.16.3

Because the royal flush is statistically much harder to achieve, it holds a higher position in the poker hands ranked hierarchy.423