In lectures we have seen that the set {1, 2, . . . , n} has 2n subsets. Now say that a set is spaced-out if it does not contain two consecutive numbers. For example, {1, 2, 3} has five spaced-out subsets: ∅, {1}, {2}, {3}, {1, 3}. How many spaced-out subsets does {1, 2, 3, 4} have? Can you guess how many spaced-out subsets {1, 2, . . . , n} has? Can you prove it?

Suppose that you had infinitely many people, each one wearing a uniquely numbered button: 1, 2, 3, 4, 5, .... You also have lots of pennies. You give each person a penny; then ask everyone to flip his or her penny at the same time. Then you ask them to shout out in order what they flipped (H for heads and T for tails). So you might hear: HHTHHTTТТHTTHTHТHTHHНННТН... or THTTTTHTHHHTTHTHTHTTTTTHHT... and so forth. Consider the set of all possible outcomes of their flipping (all possible sequences of H's and T's). Does the set of possible outcomes have the same cardinality as the natural numbers? Justify your answer.

A set contains fourteen elements. How many subsets can be formed from this​ set?

Consider numbers of the form a = a₁ª2a3….. an, with n digits, each digit from the set {1, 2, 3, 4, 5, 6, 7, 8}. For example, if n = 4, the number a = a₁ª2ª3a4 will have 4 digits from the set {1,2, 3, 4, 5, 6, 7, 8}. Let n = 7. a. How many such numbers are there? b. How many c. How many such numbers are there for which the sum of the digits is even? such numbers contain more even digits than odd digits?