I am trying to use these two exercises to study for my exam. I plan on offering

Question

I am trying to use these two exercises to study for my exam. I plan on offering the maximum number of points in the hopes that I can get a very detailed answer I will be able to use to completely understand the problems.

If I can get an answer within the next couple of hours I would be willing to gift some extra points as well.

Thanks!

How many subsets of the set {1, 2, ..., 100} have more than one element? How many bitstrings s of length 10 satisfy at least one of the following two properties: s has at least 3 consecutive 1s. s has at least 4 consecutive 0s.

Explanation / Answer

There are 2^100 - 1 subsets of the given set. 100 of them have just one element. Your answer: 2^100 - 101

Take 3 consecutive zeroes as one element and take x 1s and 7-x zeroes. Permute and add all possibilities.

Do the same for 4 consecutive ones.

Add the two.

Take 3 consecutive 1s and 4 consecutive zeroes, 3-x 1s and x 0s. Subtract from the previous to eliminate repetitions.

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