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Statistics 9-12 / Week 11 / Tuesday
2/6
Week 11 · Counting and the Multiplication Rule

Tuesday

Orders and groups
// Volunteer shifts drawn two at a time
⏱ about 20 min

Tuesday: Orders and Groups

"6 volunteer cards, 3 one-hour shifts in a row," Wren says. "How many ways can the morning fill up?"

"6 for the first hour, 5 for the second, 4 for the third," Comet counts. "6 × 5 × 4 = 120."

"That counts shift orders," Wren says. "A then B then C is different from C then B then A."

"But they are the same three people," Comet says. "Same morning crew. What can we make of that?"

Nova projects three cards shuffling among three hours. "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. The shifts are in time order, so shift orders are permutations.

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

All 3 shifts with 3 people: 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 volunteers work the morning is a combination.

Every group of 3 appears 6 times among the 120 orders, once per time 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 timed shifts from 6 volunteersyesP(6, 3)120
Choose 3 volunteers for the morning from 6noC(6, 3)20
Arrange 3 people in 3 shiftsyesP(3, 3) = 3!6
Order 4 pacers for the relayyesP(4, 4) = 4!24
Pick 2 tokens from 8, order ignorednoC(8, 2)28

From counts to probabilities

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

Cards A and B both on the morning shift: the group holds both, plus one of the other 4. That is C(4, 1) = 4 groups.

P(A and B together) = 4/20 = 1/5. One group in five puts them together.

COUNT ORDERS AND GROUPS
  • Read the question.
  • Tap your answer.
6 volunteers, 3 one-hour shifts in time order, one person per shift. How many shift orders are possible?
From 6 volunteers, how many different groups of 3 can work the morning?
The 4 pacers run the relay one after another. How many running orders are possible?
From 5 volunteers, how many different pairs can staff the water table?
COUNTS INTO PROBABILITIES
  • Read the question.
  • Tap your answer.
One morning group of 3 is drawn by lot from 6 cards. What is the probability cards A and B are both in it?
A time order is drawn by lot for one group of 3. What is the probability card A gets the first hour?
Which question is answered by a combination, not a permutation?
COUNT THE MORNING GROUPS C(6, 3), IN ORDER
  • ?Notice every group was counted 6 times
  • ?Read it back: 20 different morning groups of 3
  • ?Count shift orders: 6 × 5 × 4 = 120
  • ?Divide: 120 ÷ 6 = 20
  • ?Count the time orders of one group: 3! = 6
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 morning group of three is a permutation.?
When all groups are equally likely, a probability is favorable groups over all groups.?
WHY THIS EXERCISEThe volunteer draw uses both counts: groups for who works together, orders for who takes which hour.
Try it
Write four letters 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 double draws, a tally, and a check against the rule.

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