Not so rigorous
How many set of cardinality do there exist such that the sum of any elements of is prime number?
The answer is 0.
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1 solution
See, 1 7 = 4 + 4 + 4 + 4 + 1 . We are told to take any 5 elements. Also, 5 has 5 residues, 0 , 1 , 2 , 3 , 4 and 5 ∣ 0 + 1 + 2 + 3 + 4 . So, in case of 1 7 if we take a number each of different residue then that will be divisible by 5 . And if we let disallow one residue then there will be 5 numbers with same residue , then the sum will be divisible by 5 . Then all are not primes.