Prime power divisors of binomial coefficients

It is known that for sufficiently large n and m and any r the binomial coefficient (ⁿ_m) which is close to the middle coefficient is divisible by p^r where p is a ‘large’ prime. We prove the exact divisibility of (ⁿ_m) by p^r for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.