S2 : ESoft Target

NUMB3RS - Season 2 - "Soft Target" - Is This Seat Taken?

Topic: Combinatorics and Permutations • Difficulty: Algebra 2

Step 1 of 6

Row Permutations & Factorials

When arranging distinct objects or people in a linear row, each decision reduces the number of available options for the next spot.

For example, if we have $3$ people sitting on a bench:

The first seat can be filled by any of the $3$ people.
The second seat can be filled by either of the remaining $2$ people.
The final seat must be filled by the $1$ remaining person.

According to the Fundamental Counting Principle, the total number of unique ways to seat them is: $3 \cdot 2 \cdot 1 = 3! = 6$

Now, let's scale this up. If we have $5$ people to seat in a row, how many unique linear seating arrangements are possible?

If there are 5 people to be seated in a row, how many different ways can they be seated?

Permutations Sandbox

Explore physical seating constraints live. Rotational symmetry, grouping restrictions, and alternating setups are represented interactively below.

ASeat 1

BSeat 2

CSeat 3

DSeat 4

ESeat 5

Active Row Seating Factorial Formula:

5! = 5 • 4 • 3 • 2 • 1 = 120

Try changing your active step on the left. The visualizer matches the respective counting principle for your current workbook question.

BakerBot Socratic Tutor

Hello! I'm BakerBot, your personal mathematical guide. I'm named after the famous codebreaker and applied mathematician. Let's work together to figure out how permutations, factorials, and round table seating setups actually work. What can I help you visualize or solve?