Learn

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

- 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. - 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*. - When
*m*different events can occur in*n*independent ways, there are*m*possible ways for all of them to occur.^{n} - 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| - One interesting pattern in Pascal’s triangle is that the sum of entries in row
*n*equals*2*^{n} - 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)
- 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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