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## Statistics and probability

### Course: Statistics and probability>Unit 8

Lesson 2: Permutations

# Factorial and counting seat arrangements

Learn how to use permutations to solve problems involving ways to arrange things. Permutations involve using factorials to count all possible arrangements. This video also explores examples including arranging three people in three seats and five people in five seats.

## Want to join the conversation?

• Okay, so this makes sense, but what's a good explanation for why we multiply instead of add, other than simply saying "because it gives us the right answer"?
• We multiply because these quantities depend on each other. If they are independent of each other we add. Hope this helps!
• So what if there are 5 people and 15 chairs?
• Switch your frame of reference - choose people for the chairs, and not chairs for the people.
• Can we have factorials for negative numbers?
• Not typically. There is a generalization of the factorial function called the gamma function, but even this doesn't give values for negative integers (though it does for all other real numbers).
• In this video Sal discuses how people can be arranged around a round table.
What if the table was in any other shape like a rectangle ? How can the number of arrangements be found then ?
• If I'm not mistaken, then you can use the same method. I don't think the shape of the table matters. But please don't depend on this answer, I'm very, very new to calculus. Just wanted to help!
• What if the number of seats is greater than the number of people or people must sit in a certain seat? Then how would you do it?
• Good question. The permutation formula works, but you need to think of it in the right way. In this instance, you can think about how many ways you can put the SEATS under the PEOPLE. If you have 5 people and 8 seats where order matters, you can put the first person in any of the 8 seats (put any of the 8 seats under person 1), the second person in any of the remaining 7 (put any of the remaining seven seats under the second person), etc. This gives 8*7*6*5*4=8!/3!= 8 P 5
• What if we have 5 students (A,B,C,D,E) and 5 chairs, but A and B refuse to sit next to each other?
• There are 5! possible seating arrangements without the condition. From that, we subtract all arrangements where A and B sit next to each other.

The easiest way to do that is to count A and B as one person. But we need to careful, because if A and B sit next to each other, the order can be AB or BA.

Possible seating arrangements of 4 people = 4!
Since A and B can be arranged as either AB or BA, possible seating arrangements where A and B sit next to each other = 2 * 4!

5! - (2 * 4!) = 72

• What about calculating 5.3! (FACTORIAL of decimal ) . .You ask google for that It gives
5.3! = 201.813275185 .
The Gamma Function can be used but is there simple explation how calculators (GOOGLE CALCULATOR) calculates it . Is there a simple algorithm ?
• Why do you multiply the 3, 2, and 1 instead of adding them?
• It is only the factorial rule that tells us to multiply. In this case, though, adding and multiplying would work the same way; we would still get 6. But to follow the rule, we must always multiply. Hope this helps!
(1 vote)
• What is something like (-2)! or even (-1 2/3)!?
Is it impossible to do this?
• This is a tricky question. Fraction factorials and decimal factorials (like (2/3)! or 0.23! ) can be solved using a pretty complicated function called the "Gamma Function". However, the Gamma Function does not apply to negative numbers.

That said, some mathematicians have attempted to "define" negative factorials. But most of the definitions for negative factorials are considered pretty useless for most purposes, and aren't mainstream. And if defining something is useless anyways, then why do it?

So the typical answer to your question is that factorials for negative numbers should just be treated as undefined. The Gamma Function treats them as undefined as well.
`five factorial = 5!`
`eight factorial = 8!`