Combinatorics Cameron -- Lucas' Theorem Proof Hi everybody -- Im currently going through Peter Cameron's combinatorics book, and I'm having trouble understanding a step in the proof of Lucas' Theorem, given on page 28 for those of you with the book. The theorem states for p prime, m = a0 + a1p + . . . + akpk n = b0 + b1p + . . . + bkpk where 0 ≤ ai, bi < p for i = 0, . . ., k -1. Then: (m choose n ) ≡ [itex]\prod[/itex] (ai choose bi ) (mod p), where the product is taken from i = 0 to i = k. The proof then states: It suffices to show that, if m = cp + a and n = dp + b, where 0 ≤ a, b < p, then (m choose n ) [itex]\equiv[/itex] (c choose d) * (a choose b) (mod p) FOR a = a0 b = b0 c = a1 + . . . + akpk-1 d = b1 + . . . + bkpk-1 I understand the proof of this statement, but I don't know why proving this is sufficient to proving the original theorem. Any help would be greatly appreciated! Thanks!