Level 5: Choosing, combinations

Counting and combinations · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 you arranged things in order. In this level the order stops mattering: you choose teams, pizzas, routes and card hands, where only who is in counts. One idea, dividing by the orders that look the same, gives the "choose" numbers \(\dbinom{n}{r}\), and they turn out to fill Pascal's triangle. The level ends with committees that come with conditions and a full count of every poker hand.

Lesson	You will be able to
Choosing, not arranging	Tell a combination from a permutation, and count combinations with \(\binom{n}{r} = {}^nP_r / r!\), explaining why the division is exact
Choosing who is in is choosing who is out	Use \(\binom{n}{r} = \binom{n}{n-r}\) to save work, find the biggest choose number, and split people into named or unnamed groups
Pascal's triangle	Build the triangle, prove Pascal's rule by asking "is this person in or out?", and explain why row \(n\) adds to \(2^n\)
Paths on a grid	Count shortest routes on a grid by choosing which steps go north, including routes through or around a point or a closed street
Committees with conditions	Count groups with "exactly", "at least", special roles and people who refuse to work together, and avoid the overcounting trap
Cards and hands	Count 5-card hands, four of a kind, full houses and every other poker hand, and check that they add up to \(\binom{52}{5}\)

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