Please if you could show the work for each problem and also

Question

1. Please if you could show the work for each problem and also if you could advise the theorem used to solved the problem.
2. Does n2 + n +11 always represent a prime number for all nonnegative values of n?
3. How many 5 sequences on {1,K,10} are there?
4. How many 5 permutations on {1,K,10} are there?
5. How many 5 permutations on {1,K,10} that begin with 2 are there?
6. In MANYDOLLARS Lotto, a subset of 7 numbers is selected at random from {1,K, 52} as the winning numbers. Given that a particular lottery drawing results in the set {a, b, c, d , e, f , g} where the letters a, b, c, d, e, f, and g represent the seven winning numbers for this drawing,
   1. How many subsets of {1,K, 52} contain exactly one winning number?
   2. How many subsets of {1,K, 52} contain exactly two winning number?
   3. How many subsets of {1,K, 52} contain exactly five winning number?