In the Primes in P paper there is a section which states that there are at least $t+l \choose t-1$ distinct polynomials of degree $<t$ in a certain group. I think this can be deduced only from the line above that states that the $X+a$ are unique and non zero. My idea: There are at least $l+1 \choose i$ unique polynomials of degree $i$ (just choose $i$ of the $l+1$ different $X+a$) and therefore there must be at least $\sum_{i=1}^{t-1} {l+1 \choose i}$ many polynomials of degree $<t$. My hope was that $\sum_{i=1}^{t-1} {l+1 \choose i} = {t+l \choose t-1}$ but I don't know if this is even true? I tried using Pascal's identity, which at least for me did not work.
Any suggestions on the last equality or on the original statement are highly appreciated.
I am highly confident that this question is self contained and could also be asked in the following way: Given l+1 distinct polynomials of degree one show that there are at least $ t+l \choose t-1$ polynomials of degree $<t$.
Nevertheless I included the link to the "PRIMES is in P" paper https://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf as there is still a possibility that I am mistaken.