Factorial!

FEBRUARY 19TH, 2017

J: John and Paul must stand side-by-side for a concept photo.  How many ways can the photographer arrange them?

S: 2.

J: What are those ways?

S: John on the left and Paul on the right, and Paul on the left and John on the right.

J: George joins them.

S: Now how many ways?

J: NOW how many ways?

S: 6.

J: Ringo shows up.  Now how many?

S: 24.

S: When there were 2 Beatles, there were 2 ways; when there were 3 Beatles, there were 6 ways; and when there were 4 Beatles, there were 24 ways.  2 · 3 = 6, and 6 · 4 = 24!  A PATTERN!!!

J: So, Pete Best wanders by.

S: But if Ringo’s there, he’s already been kicked out.

J: Yes, but these nice guys will let him be in the photo.

S: They won’t.

J: They’re getting paid £10,000,000 to do it.

S: Okay, fine.

J: So, how many Beatles are there?

S: 4, because Pete isn’t a Beatle anymore.

J: How many Beatles, new and old, are there?

S: 5.

J: And how many ways can the photographer arrange them?

S: 24 · 5 = 120.  So, the photographer can arrange them 120 different ways.

J: How many places are there where Pete could stand?

S: 5.

J: And, for each place Pete stands, how many ways can the other four Beatles be arranged?

S: 24.

J: So, that’s how you get the 24 · 5 = 120.  This 120 is 5 · 4 · 3 · 2.  This thing has a name: factorial.

S: FAAAAAAAAAAAAACTOOOOOOOOORIIIIIIIIIIIIIIIIIAAAAAAAAAAAAAAAAAAL!!!

J: So, we could write it as 5! (five followed by an exclamation point).

S: So, the exclamation point is also a math symbol?

J: Mm-hmm.  Yes.

S: That’s weird.

J: Mm-hmm.

S: Okay.  Go on.

J: Stu shows up.

S: YAAAAAY!  STU!!!

J: How many ways can the photographer arrange them now?

S: 2 · 3 · 4 · 5 · 6, or 6!, otherwise known as 120 · 6, otherwise known as 720.  720 DIFFERENT POSSIBLE PHOTOS.

J: For just 6 people!

S: People! Get it?

J: Hahahaha!

S: People factorial!!!  That’s what you said.  Wait – is this the end of the lesson?

J: Well, what if Brian and Linda show up, AND the other George – what’s his name?

S: George MARTIN!!!  How could you not know?!

J: So, how many people must be arranged in the photo?

S: This doesn’t work.  Stu died long before Paul met Linda!

J: She just happened to wander by!

S: But they didn’t know her!

J: But they needed a woman in the photo!

S: But she lived in New York!

J: And they were doing the photo shoot in New York!

S: It still isn’t realistic.

J: Then who?  Who’s their most famous fangirl?

S: There is no “most famous” fangirl!

J: How many hung out at Apple?

S: I don’t know – a lot – but this is in New York, remember?

J: Weren’t there four that were in the book – George wrote a thank-you to one of them – something like that?

S: Yes, but they’re in New York!  How about Frieda Kelly?  She was their secretary and she started out as a Beatles fangirl in the Cavern days.

J: Okay.  So, how many people must be arranged in the photo?

S: Wait – what  what about Mal and and Neil?

J: Okay.  They get in the photo, too.

S: Now, there are 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · 11 different ways for the photo to be arranged.  I’m getting a calculator.  Okay.  There are 39,916,800 ways, or 11!.  That photo shoot would take years, and everyone would start fighting with each other.

J: Now, all thousand school chums show up and join them.

S: Like Ivan?

J: Mm-hmm.

S: Do you even know who Ivan is?

J: I remember the name.  And surely, out of one thousand school chums, they had at least one named Ivan.

S: IVAN WAS THE MOST IMPORTANT PERSON IN BEATLE HISTORY!  HE WAS THE ONE WHO INTRODUCED JOHN AND PAUL!!!  So, without him, we wouldn’t be doing Beatles math right now.

J: Oh.  So of course he has to be in the photo!  So, how many people will be in the photo?

S: 1,009.

J: And how many ways must they be arranged?

S: Um…um…something factorial!

J: Yes.  What number factorial?

S: Well, not 1,009, because when there were 9 people, it was 11!

J: No!  I thought there were 11 people for 11 factorial!  No, no, no…  There’s been a mistake.  I thought we were going from 11 people to 1,000!  So, I don’t know where the 9 came from…

S: Let’s review it…  Oh!  It WAS 11!  I forgot about Mal and Neil!  So, 1,011! (or one thousand eleven factorial).  NOW are we done?

J: Not yet.

S: NOT YET?

J: No, but I won’t make you type in 1,011 multiplications on a calculator.  So, I want to say something: For a lot of school calculators, 69! is the highest the calculator can handle.

S: And on mine, it doesn’t even have factorial!  Okay, I looked up 1,011! on your phone calculator, and it is 4.297327055E2600.  NOT A REAL NUMBER!  I LOATHE WHEN CALCULATORS GIVE “E” AS AN ANSWER!!!

J: It actually is a real number, as opposed to a natural number.  Oh, I misspoke.  So, it’s clearly kind of fuzzy, right?  But it’s not very satisfying, is it?

S: It’s not.  Which is why I loathe “E”.

J: That’s a pretty big number.

S: Bigger than undecillion?

J: Yes.

S: WOW.

J: So, let’s say that all the people are fast, and they can rearrange themselves for each photo in just one second.  So, how long will it take to take all the photos?

S: 4.297327055E2600 seconds.

J: Uh… How many minutes?

S: 4.297327055E2600 ÷ 60.

J: Well, there are 86,400 seconds in a day.  And 365 days in a year.

S: Let me do 86,400 · 365… Oh!  It’s only 31,536,000!  Uh-oh.  That means they’ll be taking photos for a LOOOOOOOOONG time.  What are you doing with my calculator?

J: Just a silly little exercise.  So, how many years will it take to take all the photos?

S: Another dumb “E”!  1.355205060376327403635…

J: Wait, you multiplied!  We want to divide by that.  We divide by the number of seconds per year to find out how many years.

S: Another dumb “E”!  What’s the difference?

J: Indeed.  How old is the universe?

S: Around 13,000,000,000 years old.

J: Sure.  So, is it practical to take photos of all those arrangements?

S: No.

J: Correct.

S: Is that the end of the lesson?

J: Tiny little wrap-up: For 4 Beatles, we had 4! arrangements.  For 3 Beatles, we had 3! arrangements.  When there were just John and Paul, we had 2! arrangements.

S: But not even 2!!  It was just plain old 2.  Please tell me we’re done.

J: Almost.  So, can you picture the different arrangements – the different ways to arrange John and Paul?

S: Yes.

J: So that’s HOW many ways?  2 factorial.  2! ways to arrange them.

S: That’s a nice ending, isn’t it?  HINT HINT

J: Almost.  Now, there are two more photos to take.  John is there.  So, how many Beatles are there in the photo?

S: 1.

J: So, how many ways can that one Beatle be arranged?

S: 1. Or 1!.

J: And how much is 1 factorial?

S: 1.

J: John has grown bored of this, and wanders off to catch up with Paul.  How many Beatles are there to arrange?

S: 0.

J: How many ways can the photographer arrange the Beatles?

S: 0!  We’re done!

J: Almost.  So, the photo is taken, so that’s an arrangement.  The photo is taken with 0 Beatles.  So, if we’re counting, how many ways can we arrange 0 Beatles?

S: 0 factorial.

J: How many photos do we have of 0 Beatles?

S: 1.

J: So, 0! is equal to…?

S: 1.  NOT 0.  That’s weird.

J: That’s a special case for the convenience of mathematics working out for us.

S: LESSON CLOSED.

J: Good job!

S: THE END.