Today, I took this observation from my note book. I am looking the strategy to deal this statement. The difference between $\binom{n}{p}$ and $\left\lfloor\frac{n}{p}\right\rfloor\,$ is divisible by p for a positive integer n and p is prime with >1. Here $\binom{n}{p}$ is the number of ways one can choose p out of n elements and $\left\lfloor{x}\right\rfloor\,$ is the greatest integers not exceeding the real number x. The above one is I found from the following problem.
5 divides the difference between $\binom{n}{5}$and$\left\lfloor\frac{n}{5}\right\rfloor\,$
Numerically we can solve. I would like to learn how to solve or prove the above cited statement mathematically?
Thank you.
I got good reply from one of the MATH STACK USER. I studied as per his guidance about the LUCAS Theorem, I encounter the following facts with doubts and difficulties.
If we express the p (not prime) in terms of $q^x$ k where q and k are relatively primes with q is prime,. Then my example given above fails. Of course x and k are not equal to 1 simultaneously. With reference to the above fact, how we generalize the above fact mathematically? Now, my second doubt/question is, why to solve my statement by Lucas Theorem? If we can do the same by Wilson’s theorem? This is I am just guessing. I am not sure how far I am correct. Kindly discuss, if I am wrong/correct? If Lucas Theorem only will solve my statement, how to encounter the fgollowing fact from Lucas theorem? For a and q are positive integers and greater than 1, such that $\binom{na}{ma}$ $\equiv 3\ $$\binom{n}{m}$ (mod p) For every pair of integers n greater than equal to m greater than equal to 0 with a & q are powers of the same prime p ? I am so exited to encounter the above facts during my study on Lucas theorem to complete my statement given above. Kindly discus and thank you so much for every replier.