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Here’s a summary of what we’ve learned:

  1. If there are p ways to do one thing, and q ways to do another independent thing, then there are p * q ways to do both things.
  2. If there are p ways to do one thing and, distinct from them, q ways to do another thing, then the number of ways to do p or q is p + q.
  3. When m different events can occur in n independent ways, there are mn possible ways for all of them to occur.
  4. We can compute the number of elements that satisfy at least one of several properties. For two sets A and B, |A ∪ B| = |A| + |B| − |A ⋂ B|
  5. One interesting pattern in Pascal’s triangle is that the sum of entries in row n equals 2n
  6. Another interesting pattern in Pascal’s triangle is that when you descend diagonally at first, then slant to form the shape of a hockey stick, the number in the slanted portion is the sum of the numbers that descended diagonally (hockey stick pattern)
  7. Counting theory has many applications in computer science. It can be applied to count bitstrings, or to find valid passwords based on password rules.

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