10 random people gather in a room. A researcher is inquiring if any two share a birthday (month and day). None of the individuals were born in a leap year.
The probability that none of the 10 individuals share a birthday (month and day) $=\left(\frac{1}{365}\right)^{10}\left(\frac{365 !}{355 !}\right)$
Why is the numerator of the second term greater its denominator?
Because the specific expression,
$$\left(\frac{1}{365}\right)^{10}\left(\frac{365 !}{355 !}\right)$$
is nothing more than syntactic sugar for the probability of consecutive dependent events, which can be more descriptively (but equivalently) expressed as
$$\frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{356}{365}. \tag1 $$
Notice, that in (1) above, the numerator of the 2nd fraction has changed from $(365)$ to $(364)$. This change accommodates that the 2nd person must have a birthdate different from the first person.
Similarly, in (1) above, the numerator of the 3rd fraction has descended to $(363)$. This descent accommodates that the 3rd person must not share a birthday with either of the first two people, and the first two people must have been born on different days of the year.
So, in (1) above, the pattern of descending numerators is justified.