Let n=51!+1 then the number of primes among n+1,n+2,....n+50

First you need to review the definition of the factorial: m!=∏i=1mi=1×2×3×⋯×m. This means that m! is divisible by 2, by 3, by 4, by every number up to m. Therefore 51! is divisible by 2, by 3, by 4 and by every number up to 51 (and a few others greater than 51, but you don't need to worry about those for this problem). Then 51!+2 is also divisible by 2. 51!+3 is also divisible by 3. 51!+4 is also divisible by 4. And so on and so forth to 51!+51, which is divisible by 51. Maybe 51!+1 is prime. Maybe so is 51!+53. But in .between those two numbers, there are zero primes.