TwiddleLab / Math / Counting and Patterns

Choosing a Team

Lesson 3 of 11 · 7 min · Free, no account needed

What you will learn

• When order does not matter, it is a combination: a team, a pair, a handful.
• Count the ordered picks, then divide by the orders of each group: 5 × 4 × 3 ÷ (3 × 2 × 1) = 10.
• We write it C(n, k) and say "n choose k". Pairs from n people: n × (n - 1) ÷ 2.
• Choosing 3 to play is the same as choosing 2 to sit out, so C(5, 3) = C(5, 2).

Start the lesson

How the lesson goes

1. A squad of 5. The coach needs a team of 3.

   How many different teams can the coach pick? Drag to your bet.

2. Pick your team: tap three players, one after another.

   Then watch every order you could have picked them in.

3. Sort these pick orders by team.

   Drag each card into the box of the team it makes.

4. A group where order doesn't matter is a combination.

   Count ordered picks, then divide by the ways to order each group.

   Teams of 3 from 5: 5 × 4 × 3 ÷ (3 × 2 × 1) = 10.

   We write it C(5, 3) and say "5 choose 3". This is a combination.

5. Game: collect all 10 teams for the wall.

   Select three players and sign them. A repeat costs a star.

6. 5 people meet. Everyone shakes hands with everyone once.

   Tap two people to draw their handshake.

7. 8 people shake hands, everyone with everyone, once. How many handshakes?

   A handshake is a pair of people: 8 × 7 ordered pairs, each pair counted twice, so 56 ÷ 2 = 28.

8. A simple sort compares every pair of cards once.

   10 cards. The orange one takes part in 9 comparisons. How many comparisons in all? Tap a total, then Check.

   Each comparison is a pair of cards, so the count is 10 choose 2 = 10 × 9 ÷ 2 = 45.

9. Orders grow. Teams rise and fall.

   Slide k, the number of players chosen, and compare the two bars.

Previous lesson: Arrangements: Factorials as Trees · Next lesson: Pascal's Triangle · Part of the course: Counting and Patterns

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