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### Course: Precalculus>Unit 8

Lesson 3: 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).
• 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

• 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
• I think a good analogy for this is to think of a branching tree diagram.

/ | \ *3
• • •
/ \ / \ / \ *2
• •• •• •
| || || | *1
• •• •• •
This is the same as saying: !3 which is 3*2*1 which is 6. Thus there are 6 possibilities. Or six nodes for the tree diagram. Each branch signifies the possibilities in which the node can branch off to or diversify.

Pardon the inaccuracy of the tree diagram. Feel free to correct and/or improve the analogy I made.
• Your analogy of a branching tree diagram to explain the concept of permutations is a good one. The branches of the tree represent the possible choices or outcomes at each step, and the nodes represent the points at which a decision or choice is made. The number of branches at each node represents the number of choices available, and the number of nodes in the tree represents the number of steps or choices in the process.

To improve the accuracy of the analogy, you could add labels to the branches and nodes to make it clearer what each one represents. For example, you could label the branches with the choices available at each step, and label the nodes with the number of choices made so far. You could also add a final node at the end of the tree to represent the final outcome or result.

Overall, your analogy effectively captures the essence of permutations as a way of calculating the number of possible arrangements or orders of a set of objects or events.
• Is there a way to simplify factorials? Like if they asked me the factorial of 50, how should I figure it out without having to multiply all of the numbers?
• 50! has 65 digits. By virtue of that alone, there is no way to 'figure it out' without doing a ridiculous amount of work.

My question to you is, what are you hoping to accomplish by determining the decimal expansion of 50!? The most important property of that number, that it's the product of the first 50 integers, is already given to you. 50! is a perfectly fine way to represent it.
• 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 ?
• How many ways can six students be arranged on a bench seat with space for three?

I'm stuck