Given $ p = \binom{2562}{2019} + \binom{2562}{2021} + \binom{2562}{2023} + \binom{2562}{2025} +.... +\binom{2562}{2561}$
and $q =\binom{p+2562}{2019} + \binom{p+2562}{2021} + \binom{p+2562}{2023} + \binom{p+2562}{2025} +.... +\binom{p+2562}{2561}$
Find :
(i) $\ \binom{p}{q}+\binom{q}{p}$ modulo by $2019$
(ii) $p+q$ modulo by $2019$
First of all I thinking of Lucas's Theorem ,but it seems only apply to $p$ .
What about $q$ ? It's like we working on a ton of digits.I can't solve anymore.
Some of my friends said that $p=q$ .Why do they thinking or I'm missing to interpret $p$ and $q$ ?
What are you thinking about this problem? Please tell me if it has a flaw in problem or give me some hint to modulo them (espacially $q$). Thank you and I appreciate for any helps.
Help(not an answer)
As $p+l!=p!\cdot l!\cdot {{p+l}\choose{p}}$ each term is a scaled version of the terms for $p$ by $p!\cdot {{p+l}\choose {p}}= {{p+l}!\over l!}$