A note on n! modulo p

Let p be a prime, epsilon > 0 and 0 < L + 1 < L + N < p. We prove that if p(1/2+epsilon) < N < p(1-epsilon), then #{n! (mod p); L + 1 <= n <= L + N} > c( N log N)(1/2), c = c(epsilon) > 0. We use this bound to show that any lambda not equivalent to 0 (mod p) can be represented in the form lambda = n(1)!...n(7)! (mod p), where n(i) = o(p(11/12)). This refines the previously known range for n(i).