Investigate!
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The Stanley Cup is decided in a best of 7 tournament between two teams. In how many ways can your team win? Let's answer this question two ways:
How many of the 7 games does your team need to win? How many ways can this happen?
What if the tournament goes all 7 games? So you win the last game. How many ways can the first 6 games go down?
What if the tournament goes just 6 games? How many ways can this happen? What about 5 games? 4 games?
What are the two different ways to compute the number of ways your team can win? Write down an equation involving binomial coefficients (that is, \({n \choose k}\)'s). What pattern in Pascal's triangle is this an example of?
Generalize. What if the rules changed and you played a best of \(9\) tournament (5 wins required)? What if you played an \(n\) game tournament with \(k\) wins required to be named champion?