Comment by ColinWright

    Placing only 23 keys into a hash table with 365 entries with
    a perfectly random hash function will probably result in at
    least one collision.

No, it won't "probably" result in at least one collision, it will result in at least one collision with probability about 1/2. As quoted elsewhere, placing k objects in a hash table with N entries will result in no collisions with probability about exp(-(k^2)/(2N)).