Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.

How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime?
1. **Understanding the problem**: We need to find how many different values are taken by \( a_j \) if all the numbers \( a_j \) (for \( 1 \leq j \leq n \)) and \( P \) are prime. Here, \( P \) is the sum of all \( p_k \) with odd values of \( k \) less than or equal to \( n \), where \( p_k \) is the sum of the products of all possible combinations of \( k \) of the numbers \( a_1, a_2, \ldots, a_n \).

2. **Case \( n = 3 \)**: For \( n = 3 \), we have:
   \[
   P(a_1, a_2, a_3) = a_1a_2a_3 + a_1 + a_2 + a_3
   \]
   Since \( P \) is a prime number greater than 2, it must be odd. This implies that the sum \( a_1 + a_2 + a_3 \) must be odd (since the product \( a_1a_2a_3 \) is always odd if all \( a_i \) are primes).

3. **Considering the parity of \( a_i \)**: For \( P \) to be odd, the sum \( a_1 + a_2 + a_3 \) must be odd. This requires that exactly one of \( a_1, a_2, a_3 \) is even (since the sum of three odd numbers is odd, and the sum of two odd numbers and one even number is odd).

4. **Prime numbers**: The only even prime number is 2. Therefore, one of \( a_1, a_2, a_3 \) must be 2, and the other two must be odd primes.

5. **Generalizing for \( n \)**: For any \( n \), if \( n \) is odd, then \( P \) will be the sum of products of an odd number of \( a_i \)'s. For \( P \) to be prime and odd, the sum of the \( a_i \)'s must be odd. This implies that there must be an odd number of even primes (which is only possible if there is exactly one even prime, i.e., 2).

6. **Conclusion**: Since there must be exactly one even prime (2) and the rest must be odd primes, the number of different values taken by \( a_j \) is 2 (one even prime and one odd prime).

The final answer is \( \boxed{ 2 } \).