# Permutations, Combinations & Probability (16 Word Problems)

“And the great useful model, after compound interest, is the elementary math of permutations and combinations.” – Charlie Munger.

What did Munger mean by this quote?

## Permutations and Combinations

Charlie Munger, the long-term business partner of Warren Buffett at Berkshire Hathaway, is known for his emphasis on mental models—frameworks for understanding and interpreting the world. When Munger references “the elementary math of permutations and combinations,” he’s highlighting the importance of these concepts as fundamental tools for decision-making in business and investing.

The permutations part of his statement relates to understanding how events or actions can be ordered or sequenced, which can greatly impact the outcome of business strategies or investment decisions. For example, the order in which you acquire companies, invest in assets, or execute a strategic plan can significantly change the final result.

The combinations part relates to the different ways elements can be selected or grouped without regard to order. For example, if you’re deciding how to allocate capital among several potential investments, the different combinations of investments you could choose would each have different risk and return profiles.

By mastering these concepts and applying them correctly, you can better predict outcomes, estimate probabilities, and make more informed decisions. This, in turn, can help you compound your returns over time—another concept Munger famously emphasizes. Munger recommends that anyone interested in business or investing fully understand these basic mathematical principles because they benefit decision-making exponentially.

## Examples

Here are examples of Permutations, Combinations & Probability used in trading, business, and investing.

2. Combinations in Business: Businesses often use combinations to analyze possible team structures, product mix decisions, or marketing strategies. For example, suppose a manager can select five members from a team of 10 for a special project. The number of possible ways to select the team is large. The manager can use this to calculate the number of potential teams that can be formed and further evaluate which combination could best fit.The formula to calculate combinations is:C(n, k) = n! / [k!(n-k)!]Here, n is the total number of items (or, in this case, team members), and k is the number of items to choose (or team members to select).

If a manager can select five members from a team of 10 for a special project, you would input n=10 and k=5 into the formula:

C(10, 5) = 10! / [5!(10-5)!] = 10! / [5! * 5!]

If you calculate it further, you find:

C(10, 5) = (109876) / (54321) = 252

So, the manager can form 252 potential teams.

3. Probability in Investing: Probability is a key component of investing. In finance, probability is often used to model the likelihood of various outcomes under uncertainty. For example, consider an investor who wants to assess the risk of a certain stock. They can use historical data to calculate the probability that the stock’s price will decrease by a certain percentage over a given period. For example, if the stock has decreased in price by more than 5% in a week 20 times out of 100, the investor might estimate the probability of a weekly drop greater than 5% to be 20%. This kind of probability assessment can help investors make informed decisions about risk management.

These concepts—permutations, combinations, and probability—are key tools in trading, business, and investing and are commonly used to analyze and make decisions under conditions of uncertainty.

## 16 Word Problems

Here are 16 word problems that involve permutations, combinations, and probability relevant to business, investing, success, and trading:

Permutations:

1. Business: A product development team is working on a series of updates for their existing software. They have developed five distinct new features, all of which will be introduced over the next five update cycles. However, the order in which these features are introduced can significantly impact user experience and adoption rates. Given this, the team wants to understand all possible sequences in which they can roll out these features. How many different permutations, or orders of feature introductions, are there for the five upcoming software updates?
2. Investing: An investor is looking at a portfolio containing four different bonds. In how many different orders can these bonds mature?
3. Success: A motivational speaker has six key topics she discusses during her talks. She always starts with ‘Goal Setting’ and ends with ‘Taking Action.’ How many different orders can she present the remaining four topics?

Combinations:

1. Business: A marketing team has a budget to run seven advertising campaigns but can only choose three due to resource constraints. How many different combinations of campaigns can they run?
2. Investing: An investor wants to invest in a portfolio of five stocks chosen from a shortlist of ten. How many different portfolios can be created?
3. Success: A graduate has the option of reading 5 out of 10 recommended books on career development. How many different combinations of books can the graduate choose?
4. Trading: A trader has enough capital to trade in 4 different currency pairs from a list of 9 options. How many different combinations of currency pairs can the trader choose to trade?

Probability:

1. Business: A company has found that 60% of customers buy a product after a free trial. If the company offers free trials to 10 new customers, what is the probability that exactly seven will make a purchase?
2. Investing: Based on past performance, there is a 70% chance that a particular stock will increase in value over the year. What is the probability that the stock will increase in value in exactly 3 out of 5 years?
3. Success: Studies suggest that successful entrepreneurs typically have a 20% chance of success with any given startup. If an entrepreneur starts five companies over their career, what is the probability that exactly two will be successful?
4. Trading: A trader’s strategy is successful 60% of the time. If the trader makes ten trades, what is the probability that exactly six will be successful?

Applications:

1. Business: A store has found that 10% of customers will return an item after purchase. The store sold 100 items this week. What is the probability that more than nine items will be returned?
2. Investing: An investor’s portfolio contains ten stocks. Each stock has an independent 30% chance of decreasing value over a year. What is the probability that more than two stocks will decrease in value over the next year?
3. Success: A student has to pass eight exams to graduate. They have an 80% chance of passing each of their ten exams. What is the probability that they pass all the exams on their first try?
4. Trading: A trading strategy has a 75% win rate. If the trader makes 20 trades in a month, what is the probability that he will make 15 or more successful trades?

## Key Takeaways

• The mathematics of sequences, choices, and likelihoods, also known as permutations, combinations, and probability, are fundamental principles in business, investing, success, and trading.
• Permutations give insights into how tasks or events can be sequenced, significantly affecting the outcomes in strategic planning and order of operations.
• Combinations allow us to comprehend the possibilities of selecting from a larger group. This can impact portfolio construction, advertising campaign selection, and decision-making processes in general.
• Probability serves as the bedrock for risk assessment and prediction under uncertain circumstances. It aids in quantifying potential outcomes, determining success rates, and underlining risk factors.
• The 16 problem scenarios highlighted the practical application of these mathematical principles, helping to illustrate their usefulness in real-world decision-making.

## Conclusion

In conclusion, the principles of permutations, combinations, and probability underscore the significance of decision-making, strategy development, and risk management in various facets of life, such as business, investing, personal success, and trading. They provide mathematical frameworks that help illuminate the paths we can take, the choices we have at our disposal, and the chances we might succeed or fail. By understanding these concepts and applying them astutely, we can maneuver through uncertainty with greater clarity, thereby increasing our potential to achieve desired outcomes. These concepts, therefore, aren’t merely academic topics confined to classrooms but potent tools for our day-to-day life and long-term endeavors.