Permutations and combinations: Problem type 2

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

01 Worked Example Video

Every step of Permutations and combinations: Problem type 2 explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

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.

Race medals

50 athletes race. Gold, silver, bronze are awarded (no ties). In how many ways can the 3 medals be distributed?

ways =

Class officers

A class of 25 students elects President, VP, Secretary (no student holds more than one office). How many ways can the offices be filled?

ways =

Swimmers

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

ways =

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 The Rule

P(n, r) = n(n−1)(n−2)…(n−r+1) multiply the first r shrinking terms

Order matters gold ≠ silver ≠ bronze → permutation, not combination

When order matters, it’s a permutation. Multiply r shrinking factors starting from n. Gold/silver/bronze medals are different awards, so a different order is a different outcome — that’s the permutation clue.

Common slips

(1) Multiplied too many terms. P(50, 3) uses exactly 3 factors (50 × 49 × 48), not 4 or 47. (2) Confused permutation with combination. If order doesn’t matter (choosing a committee), it’s combination. If order matters (assigning medals), it’s permutation. (3) Repeated the same person. “No ties” means shrinking pool — subtract one each step.

04 Practice — Step by Step

Practice this problem step by step

(a) A race has 10 runners. In how many ways can 1st, 2nd, 3rd be assigned (no ties)?
(b) A librarian arranges 4 books in a display window from a shelf of 8 titles. How many arrangements are possible?

Step 1

Part (a): pick n and r

For the race with 10 runners assigning top 3, what are n and r for P(n, r)?

Step 2

Compute P(10, 3)

Multiply the first 3 shrinking terms: 10 × 9 × 8.

P(10, 3) =

Step 3

Part (b): pick n and r

For the librarian arranging 4 books from a shelf of 8, what are n and r?

Step 4

Compute P(8, 4)

Multiply the first 4 shrinking terms: 8 × 7 × 6 × 5.

P(8, 4) =

Q33 Q35

Permutations and combinations: Problem type 2

You walked two permutations end to end.