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Precalculus 9-12 / Week 12 / Tuesday
2/6
Week 12 · Counting, Expected Value and Regatta Day

Tuesday

Orders and groups
// The heats drawn by lot
⏱ about 20 min

Tuesday: Orders and Groups

"6 boats, 3 lanes in a heat," Wren says. "How many ways can the first heat fill its lanes?"

"6 for lane 1, 5 for lane 2, 4 for lane 3," Comet counts. "6 × 5 × 4 = 120."

"That counts lane orders," Wren says. "Kestrel, Heron, Osprey is different from Osprey, Heron, Kestrel."

"But they are the same heat," Comet says. "Same three boats. What can we make of that?"

Nova projects the three boats shuffling among three lanes. "Would you like a hint? How many orders does one group have?"

"6," Wren says. "So every group was counted 6 times. 120 ÷ 6 = 20 groups."

"120 orders, 20 groups," Comet says. "Two counts for two questions."

Way one: permutations count orders

A permutation is an arrangement where order matters. Lanes are numbered, so lane orders are permutations.

P(6, 3) = 6 × 5 × 4 = 120. One factor for each lane, each one smaller than the last.

All 3 lanes of one heat, with 3 boats: P(3, 3) = 3 × 2 × 1 = 6. That is 3!, read "3 factorial".

Way two: combinations count groups

A combination is a group where order does not matter. Which three boats row together is a combination.

Every group of 3 boats appears 6 times among the 120 orders, once per lane order.

So C(6, 3) = P(6, 3) ÷ 3! = 120 ÷ 6 = 20.

Ask "does order matter?" first. Yes means permutation. No means combination, and you divide by k!.

QuestionOrder matters?CountValue
Fill 3 numbered lanes from 6 boatsyesP(6, 3)120
Choose 3 boats for a heat from 6noC(6, 3)20
Arrange 3 boats in 3 lanesyesP(3, 3) = 3!6
Choose 2 boats from 4 for a pair rownoC(4, 2)6
Fill 2 lanes from 4 boatsyesP(4, 2)12

From counts to probabilities

Heat groups are drawn by lot, so all 20 groups are equally likely. A probability is favorable groups over all groups.

Kestrel and Heron in the same heat: the group holds both, plus one of the other 4 boats. That is C(4, 1) = 4 groups.

P(same heat) = 4/20 = 1/5. One group in five puts them together.

COUNT ORDERS AND GROUPS
  • Read the question.
  • Tap your answer.
6 boats line up for 3 numbered lanes, one boat per lane. How many different lane orders are possible?
From 6 boats, how many different groups of 3 can be chosen for a heat?
The 3 boats of one heat take the 3 numbered lanes. How many lane orders are possible?
From 5 boats, how many different pairs can be chosen for a pair row?
COUNTS INTO PROBABILITIES
  • Read the question.
  • Tap your answer.
One heat of 3 is drawn by lot from 6 boats. What is the probability Kestrel and Heron are both in it?
Lane orders are drawn by lot for one heat of 3. What is the probability Kestrel gets lane 1?
Which question is answered by a combination, not a permutation?
COUNT THE HEAT GROUPS C(6, 3), IN ORDER
  • ?Count the lane orders of one group: 3! = 6
  • ?Notice every group was counted 6 times
  • ?Count lane orders: 6 × 5 × 4 = 120
  • ?Divide: 120 ÷ 6 = 20
  • ?Read it back: 20 different heats of 3 boats
WHY THIS EXERCISEA combination is a permutation with the repeats divided out. That one division is the whole difference.
An arrangement where order matters is called a this. Type one word.
A group where order does not matter is called a this. Type one word.
3! is what number? Type the number.
StatementTrue or false?
6 × 5 × 4 = 120?
C(6, 3) is bigger than P(6, 3).?
120 ÷ 6 = 20?
Choosing a heat of three boats is a permutation.?
When all groups are equally likely, a probability is favorable groups over all groups.?
WHY THIS EXERCISEThe lane draw uses both counts: groups for who rows together, orders for who gets which lane.
Try it
Write four boat names on cards. List every group of two, then every ordered pair.
Check: ordered pairs ÷ 2! should equal the number of groups.

Excellent. Tomorrow the bag comes out for real: twenty draws, a tally and a random variable in the Boathouse Lab.

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