There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks. 
The committee members have been provided with keys in such a way that every six members can open the safe, but no five members can do that.
What is the smallest possible number of locks, and how many keys are needed in that case?
1. **Define the problem and notation:**
   - Let \( C = \{p_1, p_2, \dots, p_{11}\} \) be the set of 11 committee members.
   - We need to determine the smallest number of locks \( n \) such that:
     - Any subset of 6 members can open the safe.
     - No subset of 5 members can open the safe.

2. **Determine the number of locks:**
   - For every subset of 5 members, there must be at least one lock that none of these 5 members can open. This ensures that no subset of 5 members can open the safe.
   - The number of such subsets of 5 members from 11 members is given by the binomial coefficient:
     \[
     \binom{11}{5} = \frac{11!}{5!(11-5)!} = 462
     \]
   - Therefore, there must be at least 462 locks, each corresponding to a unique subset of 5 members.

3. **Determine the number of keys:**
   - For each lock, there must be exactly 6 members who can open it (since any subset of 6 members should be able to open the safe).
   - The number of subsets of 6 members from 11 members is given by:
     \[
     \binom{11}{6} = \frac{11!}{6!(11-6)!} = 462
     \]
   - Each lock must be opened by exactly 6 members, so each lock corresponds to a unique subset of 6 members.
   - Therefore, the total number of keys needed is:
     \[
     \binom{11}{6} \times 6 = 462 \times 6 = 2772
     \]

4. **Verification:**
   - Each subset of 6 members can open the safe because they have all the keys corresponding to the locks.
   - No subset of 5 members can open the safe because there is at least one lock that none of these 5 members can open.

Hence, the smallest possible number of locks is 462, and the number of keys needed is 2772.

The final answer is \( \boxed{462} \) locks and \( \boxed{2772} \) keys.