Choosing 80 from 100

Question

First question, i hope i follow the rules accordingly

I tried choosing 80 numbers from the ${{1,...,100}}$ set in two ways:
The first is 100 choose 80:
$$\binom{100}{80} = 535983370403809682970$$

The other way is as follows: For each maximum value, choose all possible ways completing the set to 80 numbers (thus choosing the 79 numbers remaining from the set, ending in the highest number left):
$$\sum_{i=1}^{20}\binom{100-i}{79} = 535983370403809682969$$

What am i misssing? which set did i leave out?

Thank you.

Answer 1

You're missing $i = 21$ which corresponds to $80$ being the largest.