Permutations and combinations: Problem type 2

Two-part. Both parts ask "how many ordered arrangements?" — permutations. Multiply the shrinking choices.

01 Mini-Lesson Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Medals + city schedule v1

(a) 50 athletes are running a race. A gold medal goes to the winner, silver to second-place, bronze to third. No ties. In how many ways can the 3 medals be distributed?

(b) Aldo has 15 European cities he'd like to visit, but he only has time for 3 on his next vacation: one on Monday, one on Tuesday, one on Wednesday. He won't visit the same city twice. How many different schedules are possible?

Class officers + presentation order v2

(a) A class of 25 students is electing a President, Vice President, and Secretary — no student can hold more than one office. How many ways can the three offices be filled?

(b) A teacher needs to choose 4 students from a class of 20 to present their projects in order. How many different presentation orders are possible?

Race finishers + photo lineup v3

(a) 12 swimmers compete in a race. In how many ways can 1st, 2nd, and 3rd places be assigned (no ties)?

(b) A photographer wants to line up 5 of 10 family members in a row for a portrait. How many different lineups are possible?

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 How to Solve It

P(n, r) = n × (n−1) × ··· × (n−r+1)
r terms, starting at n and shrinking by one each step

Permutations: order matters, no repeats. Multiply r shrinking choices.

Confirm order matters.

(a) Gold vs silver vs bronze — yes, distinguishable. (b) Monday vs Tuesday vs Wednesday — yes, distinguishable. Both are permutations.

(a) Medals: P(50, 3).

50 choices for gold, 49 remaining for silver, 48 remaining for bronze:

P(50, 3) = 50 \times 49 \times 48 = 117,600

(b) City schedule: P(15, 3).

15 cities for Monday, 14 left for Tuesday, 13 left for Wednesday:

P(15, 3) = 15 \times 14 \times 13 = 2,730

▸ COMMON SLIPS(1) Treated this as a combination. Dividing P(50, 3) by 3! gives C(50, 3) = 19,600 — the answer for an unordered selection (a committee). The medal problem is ordered: keep P(50, 3) = 117,600. (2) Used n·r instead of shrinking choices. 50 × 3 = 150 is what you get if you forget that one athlete can't win two medals. The choices shrink: 50, 49, 48. (3) Off-by-one on r. P(50, 3) has three terms: 50, 49, 48. Not 50, 49, 48, 47 (which would be P(50, 4)).

04 Practice — Step by Step

Practice with the class-officer setup.

Three offices, 25 students.

A class of 25 elects President, Vice President, and Secretary — no one holds more than one office. How many ways can the three offices be filled?

ways =

Five-member lineup from 10.

Line up 5 of 10 family members for a photo. How many different ordered lineups?

lineups =

▸ NICE WORK

You've walked through the whole problem.

That's the move. ALEKS will give you a different version with different numbers — but the steps are the same.

Q3 Q5