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Combinatorics - the fancy word for the math of figuring out the number of different possibilities, based on the grouping and order if choosing of a set of objects

Example: 

Function 1- Permute Permutation is finding the number of groups in a set, but when order matters (For example, if there is a set of A,B,C,D,E, ABC is counted different from BAC due to arrangement) How do we get permutations?

Before we continue, let's learn a new function-- factorial

 So, the number of ways you can arrange N objects is N! But, what if you wanted to find the number of arranged groups of P, in a larger set of N? Let's take arranged groups of 2 (AB is counted as different from BA), from a set of 5 as an example.

 Therefore, for the number of arranged groups of P in a set of N objects, you must use N! until you have multiplied P numbers. Therefore, Permute(N,P) [nPp]=                       N!/(N-P)!

What if you want to count the number of different groups in a set, such that arrangement doesn't matter?

Function 2- Unordered Combination ("choose")

Isaac Newton was a scientific genius at a time when scientists were often denounced as heretics. 

He had one antidote to this problem: He had to prove his theories beyond any doubt, which required obsessive precision. 

In his biography on Newton, Mitch Stokes writes:

[Newton] could make make measurements with such precision that few philosophers accused him of fabricating his results … Using only a compass an his eye, Newton would painstakingly hunt down the source of minute measurement discrepancies of less than one one hundredth of an inch … no one else in the seventeenth century would have paused for an error twice that size. 

Newton’s work rests on the idea that the physical world works in ways that are measurable, and, therefore, as long as you measure carefully enough you could learn anything about everything. Newton was so good at this that we named the field after him: Newtonian physics. 

Scientists viewed the physical world through that single lens for another two-hundred years. 

Then we discovered quantum physics. It upended everything. 

Quantum physics showed us that parts of the physical world resisted the precision Newton obsessed over. We began to learn that physical things could behave in ways we never thought possible. Ways that defied logic. Subatomic particle movements were messy and unpredictable. Quantum theory taught us that precise measurements can’t exist because the act of measuring something changes its movement. Walter Isaacson writes about the famous Heisenberg Uncertainty Principle in his biography of Albert Einstein: 

It is impossible to know, Heisenberg declared, the precise position of a particle, such as a moving electron, and its precise momentum at the same instant. The more precisely the position of the particle is measured, the less precisely it is possible to measure its momentum. An electron does not have a definite position or path until we observe it. This is a feature of our universe, he said, not merely some defect in our observing or measuring abilities. The uncertainty principle, so simple and yet so startling, was a stake in the heart of [Newtonian] physics. It asserts that there is no objective reality—not even an objective position of a particle—outside of our observations. 

The emergence of two types of physics taught scientists to be humble. Neither version is better than the other. You just have to know which one to use in the right circumstance. Newtonian physics is deterministic. It gives you perfect answers. Quantum physics is probabilistic. It only offers hints that can roughly set you in the right direction. 

It’s a great way to think about investing, where the same distinction applies.

Coca-Cola is fighting more than a decade of declining soda sales. Its stock is at an all-time high. 

Snap has built one of the most viral products of our time. Its stock is down almost 50% in the last six months. 

Good investing is about more than backing companies that perform well. It’s about backing companies that perform well within the context of the market’s current and future expectations. 

A company’s performance can be measured precisely. Accounting is exact. Books balance out down to the penny. Calculating and planning a company’s unit economics requires precision. Low-margin businesses require a precise grasp of the business’s cost structure. Bank of America knows exactly what will happen to earnings if 0.01% of borrowers default on their credit cards. Like Newtonian physics. 

Market expectations aren’t like that. Expectations are moods, and moods, almost by definition, are the gap between what’s happening and how people interpret what’s happening. They resist all attempts to figure them out in ways that make sense. They change constantly, without warning or reason. They can be different for two companies that look identical from the outside. At best, you can measure them with probabilities. Like quantum physics. 

Not distinguishing between the two in investing is dangerous. There are things we can know and measure perfectly, and things we have to think about probabilistically, if not philosophically. They are different skills that attract different people, many of whom don’t realize the need for an alternative set of thinking. 

Take future market returns. You only need to know three things to know exactly where stocks are going next: The dividend yield, earnings growth, and the change in valuation multiples. 

The dividend yield is easy. We can measure it today. Earnings growth is a little trickier, but we can make a decent estimate. What about the change in valuation multiples? God, who knows. If I’m trying to calculate how much stocks will return between now and 2027, I need to know what kind of mood people will be in in 2027. And no reasonable person can claim to do that. I don’t know what kind of mood I’ll be in tonight, let alone 200 million strangers ten years from now. 

The S&P 500 index earned $100 in profits in the year ended February 2014. The 10-year Treasury bond yielded 2.5% at the time. The index traded for 18.5 times earnings. It earned the same $100 in profits in the year ended March 2017, and the 10-year Treasury bond yielded the same 2.5%. Yet the index traded for 23.9 times earnings. 

The difference between the two earnings multiples reflects a change in investors’ moods. People felt better about their investments in 2017 than they did in 2014. Why? No one knows. No one can measure. Maybe it was because the pain of the 2008 financial crisis wears off over time. Maybe because more people had jobs. Maybe because the doom-and-gloom forecasts sounded plausible in 2014 but have grown old since then. Or maybe – probably – some unfathomably complex mix of what happens when tens of millions of people try to predict the future based on the unique lens of their own day-to-day experiences and adjust their investments accordingly. Whatever it was, it acts like quantum physics – not only unpredictable but unmeasurable, even if we know what we’re looking at.

That’s how markets work. It’s how a lot of things work. Richard Feynman, the physicist, once said: “Nature isn’t [Newtonian] dammit. If you want to make a simulation of nature, you’d better make it quantum. 

”Albert Einstein spent a lot of his career arguing against quantum theory. It seemed contradictory to everything we thought about physics. 

“One cannot make a theory out of a lot of ‘maybes’” he once told a group of physicists. “Deep down it is wrong, even if it is empirically and logically right. ”“Einstein, I’m ashamed of you,” said physicist Paul Ehrenfest, according to Isaacson’s biography. Fellow physicists said he was being as stubborn as others had been when disputing his theory of relativity. 

Einstein eventually came around. He nominated the two physicists who devised quantum theory for the Nobel Prize in 1933. He wrote in his nomination: “I am convinced that this theory undoubtedly contains a part of the ultimate truth. ”You could, in fact, make a theory out of maybes. 

The development of quantum physics next to Newtonian physics shows how hard it is to grasp that some things can be measured and are in our control, and somethings can’t, and aren’t. It’s especially hard to grasp when those things are in the same field. 

The same dissonance affects investors. 

Accepting that investing is made up of both precise facts and theories of maybes is the hardest thing for investors to grasp. Theories of maybes can often be distilled into probabilities. But this doesn’t solve the problem, because we can’t calculate the probabilities of things we don’t know. The history of markets is, and will always be, the story of things that were unprecedented until they happened. That’s hard to accept if you have a Newtonian physics mind. And many investors do. 

Getting comfortable with maybe requires two things: Humility, and room for error. Humility that there’s a lot of stuff we can’t know, and room for error to offer the only protection against that uncertainty. It’s the only way to survive in an industry where some of the most important variables can’t be calculated, measured, or fully understood. 

Newton, interestingly, figured this out. After allegedly losing a fortune investing, he said he could “calculate the motion of heavenly bodies, but not the madness of people. ”