Definition
Boundedness of Binomial Coefficients
For and , the binomial coefficients satisfy
The lower bound is attained at and , while the largest coefficients occur in the middle of the row.
In particular, the two boundary coefficients are
There is exactly one way to choose no elements—the empty set—and exactly one way to choose all elements—the entire set.
Why the Maximum Is Central
Consecutive coefficients satisfy
This ratio is greater than one before the middle and less than one after the middle. The coefficients therefore increase towards the centre and then decrease. Moreover, the binomial theorem gives
so every individual coefficient is at most .