Latticework of Mental Models: Permutation and Combination

Here’s an interesting trivia.

If you wear different tie on the same shirt, most people will think that you’re wearing a different shirt. That’s an interesting way to multiply your options without really buying new cloths (except few new ties).

“That’s not a trivia, that’s a Jugaad.”, you might want to say. Anyways, That brings me to an equally interesting mindbender.

If you have 2 shirts (white, blue), 3 pants (black, gray and brown) and 3 different ties (pink, orange, red), in how many different ways can you get dressed? Assumption here is that getting dressed requires you to wear all three i.e. a shirt, a pant and a tie.

Using the multiplication principle we can say that there are total 2 x 3 x 3 = 18 ways to get dressed. Of course some of the dress combinations will look outright funny but our concern here is to find out all possible ways to get dressed. Moreover, today we are getting into Maths discipline and most mathematicians don’t really have whole lot of fashion sense anyways.

So that’s the simplest example of using the idea of combinations in real life. Now let’s say, for some strange reason, we were also considering the order in which you put on the cloths, i.e. it matters to us if one puts on the shirt first instead of tie.

Imagine wearing a tie first and then squeezing the shirt inside the tie, funny right? I told you mathematicians don’t care much about the dressing etiquettes 🙂

Okay, back to the same question again. In how many ways can you get dressed if the order of dressing matters?

For each of those 18 combinations, there are six ways to dress. So for a given combination of pant, shirt and tie you could go pant first, followed by the shirt and then tie. Or you could wear the shirt first, followed by the tie and finally the pant. Or you could do shirt, pant and tie. And so on.

Here the order (or the arrangement) of the objects matter. So permutation is just a fancy pants (no pun intended) definition of all possible ways of doing something. Simply put, permutations or rearrangements mean the different ways we can order or arrange a number of objects.

Combinations means the different ways we can choose a number of different objects from a group of objects where no order is involved, just the number of ways of choosing them.

For that matter a combination lock should really be called a permutation lock. The order you put the numbers in matters. A true combination lock would accept both 10-17-23 and 23-17-10 as correct.

In most simple terms, permutations and combinations (P&C) is all about counting the possible outcomes.

I am not sure about the current curriculums in schools but I learnt about P&C after 10th standard. Perhaps they are teaching these mathematical principles to kids in junior classes now.

Mathematicians get a kick out of converting simple numbers into complicated equations containing english and greek letters. So this is how a typical maths text book would describe the idea of permutations –

If one event can happen in ‘n‘ different ways, and a second event can happen independent of the first in ‘m’ different ways, the two events can happen in n x m (n multiplied by m) different ways.

To add little more clarity to the above definitions, it would help to think it this way – Permutations are for lists (order matters) and combinations are for groups (order doesn’t matter).

The Factorial (!)

Let me take help from Peter Bevelin, author of Seeking Wisdom, for explaining the idea of factorial. Bevelin writes –

We have 3 hats to choose from – one black, one white and one brown. In how many ways can we arrange them if the order white, black and brown is different from the order black, white and brown? This is the same as asking how many permutations there are with three hats, taken three at a time. We can arrange the hats in 6 ways:

1. Black-White-Brown
2. Black-Brown-White
3. White- Black-Brown
4. White-Brown-Black
5. Brown-White-Black
6. Brown-Black-White.

Another way to think about this: We have three boxes in a row where we put a different hat in each box. We can fill the first box in three ways, since we can choose between all three hats. We can then fill the second box in two ways, since we now can choose between only two hats. We can fill the third box in only one way, since we have only one hat left. This means we can fill the box in 3 x 2 x1= 6 ways.

Another way to write this is 3! [another fancy pant word] If we have n (6) boxes and can choose from all of them, there are n (6) choices. Then we are left with n-1 (5) choices for box number two, n-2 (4) choices for box number three and so on. The number of permutations of n boxes is n!. What n! [pronounced as n-Factorial] means is the product of all numbers from 1 to n.

Suppose we have a dinner in our home with 12 people sitting around a table. How many seating arrangements are possible? The first person that enters the room can choose between twelve chairs, the second between eleven chairs and so on, meaning there are 12! or 479,001,600 different seating arrangements.

The number of ways we can arrange ‘r’ objects from a group of ‘n‘ objects is called a permutation of ‘n’ objects taken ‘r’ at a time and is defined as

p(n,r) = n! / (n-r)!

A safe has 100 digits. To open the safe a burglar needs to pick the correct 3 different numbers. Is it likely? The number of permutations or ways of arranging 3 digits from 100 digits is 970,200 (100! / (100-3)!). If every permutation takes the burglar 5 seconds, all permutations are tried in 5 6 days assuming a 24-hour working day.

In how many ways can we combine 2 flavors of ice cream if we can choose from strawberry (S), vanilla (V), and chocolate (C) without repeated flavors? We can combine them in 3 ways: SV, SC, VC. VS and SV are a combination of the same ice creams. The order doesn’t matter. Vanilla on the top is the same as vanilla on the bottom.

The number of ways we can select ‘r‘objects from a group of ‘n’ objects is called a combination of n objects taken r at a time and is defined as

C(n,r) = n! / r! (n-r)!

The number of ways we can select 3 people taken from a group of 10 people is 120 i.e.

(10! / (3!) (10-3)!)

P&C In Investing

The study of permutations applies to investing in a broad sense because a good understanding of probability is sometimes necessary to make rational financial choices.

Infact, if you don’t understand P&C properly, you would have a tough time understanding probability. So P&C forms the foundation for understanding the concept of probability.

Charlie Munger, in his famous talk at USC Business School in 1994 entitled A Lesson on Elementary Worldly Wisdom, said –

Obviously, you’ve got to be able to handle numbers and quantities—basic arithmetic. And the great useful model, after compound interest, is the elementary math of permutations and combinations. And that was taught in my day in the sophomore year in high school…It’s very simple algebra. It was all worked out in the course of about one year between Pascal and Fermat. They worked it out casually in a series of letters.

It’s not that hard to learn. What is hard is to get so you use it routinely almost everyday of your life. The Fermat/Pascal system is dramatically consonant with the way that the world works. And it’s fundamental truth. So you simply have to have the technique.

If you don’t get this elementary, but mildly unnatural, mathematics of elementary probability into your repertoire, then you go through a long life like a one legged man in an asskicking contest. You’re giving a huge advantage to everybody else.

One of the advantages of a fellow like Buffett, whom I’ve worked with all these years, is that he automatically thinks in terms of decision trees and the elementary math of permutations and combinations.

Conclusion

For me, preparing tea is a combination problem. All I know is that I need to throw in the ingredients (tea, water, milk, sugar) in a pan and heat it for 5-10 minutes. But for my wife it’s a permutation problem. She insists that milk be added at the last. No wonder, nobody wants to drink my tea.

Okay now that you’re equipped with the knowledge of P&C, riddle me this. If my wife, my two kids and I go to a movie theatre, in how many different ways can four of us be seated on our four allotted seats?

Did you say 24?

That could be correct except that the real world hardly presents itself like a well defined mathematics problem. In real world, there’s always a catch.

The catch in my case is that my kids are identical twins. So for all practical purposes, assuming the kids are wearing the same attire, half of the seating arrangements will look identical. And the correct answer would be 12.

Talking about probability, the odds of having an identical twins is approximately 1 in 300 i.e. 0.33 percent. My wife and I consider ourselves very lucky 🙂

One goal to learn different mental models from multiple disciplines is to learn how problems can be transformed. Remember that painting of the old lady and young woman?

Do you see both? Once you can see both of them it’s easy to switch between them. That’s the beauty of multidisciplinary thinking. The more models you have available, faster you can turn problems into each other.

It’s completely fine to use one model to understand the idea, and another to work out the details. But life becomes difficult when we think there’s only one way to approach it.

Beware! These mental models and multidisciplinary ideas do no good sitting inside your head like artifacts in a museum for they need to be taken out and played with.

I hope these mental models help you cut through the optical illusion around you and you start seeing many young and old ladies – metaphorically 🙂

Take care and keep learning.