We cant lie -Â Data Science Interviews are TOUGH. Top tech companies ask very tough questions about probability and statistics
That’s why we put together 40 real chances There are answers to all 40 problems in our book, Ace The Data Science Interview. There are also answers to 161 other problems on SQL, Machine Learning, and Product/Business Sense. You can also practice some of these same exact questions on DataLemurs statistics interview questions section.
Combinatorics is an important topic that comes up frequently in technical interviews, especially for software engineering, data science, and quantitative roles. Knowing how to analyze counting problems and determine the number of possible combinations and permutations can help you stand out from other candidates.
In this article, we’ll cover some of the most common combinatorics questions asked during interviews We’ll explain the key concepts you need to know and provide example solutions to practice. Read on to learn strategies for solving combinatorics problems and ace your next interview!
What is Combinatorics?
A Brief Introduction
Combinatorics is a branch of mathematics focused on counting finite but complex structures and patterns, It involves determining the number of possible configurations within a finite set
Some key aspects of combinatorics include:
- Combinations – Order doesn’t matter. Finding subsets from a larger set.
- Permutations – Order does matter. Finding ordered arrangements from a set.
- The multiplication and addition principles – Ways to break down and solve complex counting problems.
- Recursion – Using functions that call themselves to solve advanced problems.
Understanding these core concepts allows you to analyze challenging counting problems through a combinatorics lens. Let’s look at some examples.
Key Combinatorics Interview Questions and Solutions
Here are some of the most frequently asked combinatorics questions during coding interviews
1. What is the difference between a combination and a permutation?
Solution: The key difference is that order matters in a permutation but does not matter in a combination.
A permutation is an ordered arrangement. For example, ABC is different than BAC.
A combination is a subset where order is irrelevant. For example, {A, B} is the same as {B, A}.
So if selecting 3 letters from the set {A, B, C, D}, there are 4 x 3 x 2 = 24 permutations but only 4 choose 3 = 4 combinations.
2. How many handshakes occur at a party with 25 people?
Solution: There are C(25,2) = 300 handshakes.
This is a combination problem since order doesn’t matter. We choose 2 people at a time from 25 people to shake hands.
Using the combination formula: C(n,k) = n! / (k! * (n-k)!)
Here n = 25 and k = 2, so C(25,2) = 25! / (2! * 23!) = 300
3. How many 5 card hands can be dealt from a standard 52 card deck?
Solution: There are C(52,5) = 2,598,960 possible 5 card hands.
This is asking how many combinations of 5 cards can be selected from 52 cards, order irrelevant.
Using the combination formula:
C(52,5) = 52! / (5! * 47!) = 2,598,960
4. You have 4 identical balls to place in 5 distinct boxes. In how many ways can this be done?
Solution: There are 5^4 = 625 ways to place 4 identical balls into 5 distinct boxes.
When items are identical but boxes are distinct, we can use the multiplication principle. There are 5 choices for the first ball, 5 for the second, 5 for the third, 5 for the fourth. So 5 x 5 x 5 x 5 = 625 total arrangements.
5. How many bit strings of length 8 contain exactly 4 ones?
Solution: We choose 4 positions for the ones, and the remaining can be either 0 or 1. So there are C(8,4) options for the 1s * 2^4 options for the remaining spots.
In total, C(8,4) * 2^4 = 70 bit strings with 4 ones.
6. How many ways can you climb a staircase with 5 steps?
Solution: This is a recursive problem, where each step can be either a single step or a double step.
Let f(n) = #ways to climb n steps.
f(n) = f(n-1) + f(n-2)
Base cases:
f(1) = 1 (only a single step)
f(2) = 2 (two single steps or one double step)
Working backwards using the recurrence:
f(5) = f(4) + f(3)
= f(3) + f(2) + f(2) + f(1)
= 3 + 2 + 1 = 6 ways
This demonstrates using recursion to systematically solve the problem.
7. There are 9 people including Alice, Bob and Carol. How many different dancing pairs can be formed?
Solution: * There are 9 people, so 9C2 = 36 pairs without restrictions.
- But Alice can only pair with 8 people excluding Bob and Carol. So 8 pairs including Alice.
- Similarly, 8 pairs including Bob, 8 pairs including Carol.
- Total pairs = 36 – 3 * 8 = 36 – 24 = 12 pairs
This demonstrates applying restrictions and combinations to enumerate possibilities.
Key Takeaways for Combinatorics Interview Questions
Here are some key strategies for mastering combinatorics problems:
- Recognize whether order matters – permutation vs combination
- Apply counting rules/formulas systematically
- Break down complex problems using multiplication and addition principles
- Use recursion to solve advanced problems elegantly
- Handle restrictions by subtracting prohibited elements from total set
With practice, these types of questions test your ability to analyze counting problems and determine an efficient solution. Use the examples and explanations above to get comfortable with combinatorics techniques.
Combinatorics comes up frequently in coding interviews, so being able to demonstrate your knowledge by walking through solutions is key. Use the solutions above to practice explaining your approach. With preparation, you’ll be set to tackle any combinatorics problem during your upcoming interviews!
20 Statistics Problems Asked By FAANGÂ & Hedge Funds
- [Facebook – Easy] How would you explain a confidence interval to someone who doesn’t know much about statistics?
- Let’s say you are doing a multiple linear regression and think there are a number of predictors that are linked. If the results of the regression are linked, how will that change the results? What would you do to solve this problem?
- [Uber – Easy] Describe p-values in laymanâs terms.
- [Easy Facebook] How would you make and test a way to compare two users’ ranked lists of favorite movies and TV shows?
- [Microsoft – Easy] Explain the statistical background behind power.
- [Twitter – Easy] Describe A/B testing. What are some common pitfalls?.
- Google – Medium: How would you use a series of coin flips to get a confidence interval?
- To use an exponential distribution with parameter λ to model the lifetime for a group of customers, let’s say you have the lifetime history (in months) of n customers. What is your best guess for λ?.
- Figure out the mean and variance of the U(a, b) uniform distribution.
- Let’s say that X is uniform (0, 1) and Y is uniform (0, 1). What do you think the minimum of X and Y will be?
- You pick n times from a uniform distribution [0, d] for Spotify – Medium. What is your best estimate of d?.
- You pick one number at random from a normally distributed random variable X ~ N(0, 1) once a day. How many days do you think it will be before you get a value of more than 2?
- [Facebook – Medium] Find the expected value of a random variable that is geometrically distributed.
- Thousand times, a coin was flipped, and 550 times it came up heads. Do you think the coin is fair? If not, explain.
- [Robinhood – Medium] Let’s say you have n whole numbers (1a…n) and want to pick a random order for them. Let i = j be an integer, and let a swap happen when i is in the jth position and the other way around. How much do you think the total number of swaps will be worth?
- [Uber – Hard] What is the mathematical difference between MLE and MAP?
- If you know the means and standard deviations of two parts of a dataset, how do you find the third? How do you find the average and standard deviation of the whole dataset? Is it possible to do this for K subsets?
- [Lyft – Hard] How do you pick a point at random from a circle with a radius of 1?
- [Two Sigma – Hard] Let’s say you keep taking samples from some i. d. random variables with a range of 0 to 1 until the sum of the variables is greater than 1. How many times do you expect to sample?.
- If you have a random Bernoulli trial generator, how do you get back a value that comes from a normal distribution?
20 Probability Interview Problems Asked By Top-Tech Companies & Wall Street
- One side of a fair coin has heads and the other side has tails. There is also an unfair coin with both sides having tails. It turns up tails all five times when you flip it, no matter which one you choose. How likely is it that you are flipping an unfair coin?
- [Easy Land] You and your friend are having fun. You two will keep flipping a coin until you see the signs HH or TH. If HH shows up first, you win. If TH shows up first, your friend wins. What is the probability of you winning?.
- [Google – Easy] What is the chance that a seven-game series will last seven games?
- Facebook makes it easy to tell if content is spam or not by having a content team that checks it. Ninety percent of them are careful raters and will mark twenty percent of the content as spam and eighty percent as not spam. However, the people who are still there are lazy raters who will label 20% of the content as spam and the other 20% as non-spam. Let us say that each piece of content is labeled separately, and that each rater If a rater has given four pieces of content the “good” label, how likely is it that they are a careful rater?
- Let’s say you draw a circle and pick two chords at random. What is the probability that those chords will intersect?.
- There is a test that can tell you for sure if you have a certain disease (Amazon Easy). There is a 1% error rate if you don’t have the disease. How likely is it that someone has the disease if they test positive?
- That’s it! There are 50 cards of 5 different colors. Each color has cards numbered between 1 to 10. You pick 2 cards at random. How likely is it that they are not the same color or in the same number?
- [Tesla – Easy] A fair six-sided die is rolled twice. What is the chance of rolling a 1 on the first roll and not a 6 on the second roll?
- [Facebook – Easy] How many rolls do you think it will take to see all six sides of a fair die?
- Three friends in Seattle all told you it was raining, but there is a 1/3 chance that each of them was lying. How do you know? How likely is it that it will rain in Seattle? Let’s say the chance of rain in Seattle on any given day is 0. 25.
- Let’s say you roll three dice one at a time. What is the chance of getting three numbers in a row that progressively gets bigger?
- “Bloomberg Medium”: Three ants are sitting on the edges of a triangle with equal sides. Each ant picks a direction at random and starts to move around the triangle’s edges. How likely is it that none of the ants will hit each other? What if there are k ants on each of the k corners of a polygon with equal sides?
- [Two Sigma – Medium] How many flips of the coin do you think it will take to get two heads in a row?
- What number of cards from a standard deck do you think you’ll draw before you see the first ace?
- […] A and B are playing a game where A has n 1 coins and B has n coins. They both flip all of their coins. How likely is it that A will have more heads than B?
- Let’s say you’re given an unfair coin that seems to lean more toward heads or tails. How can you generate fair odds using this coin?.
- [Quora – Medium] Say you have N i. i. d. draws of a normal distribution with parameters μ and Ï. How likely is it that k of those draws will be bigger than a certain value Y?
- [Spotify – Hard] A fair die is rolled n times. For each r in 1, what is the chance that the biggest number rolled is r? 6?.
- [Hard] There are n Snapchat users, and they are split into two groups, A and B. Each user in A is friends with users in B, and so on. Each person in A will pick a random person in B as their best friend, and each person in B will pick a random person in A as their best friend. People who have chosen each other are best friends with each other. How likely is it that there won’t be any best friends who are different?
- [Tesla – Hard] Let’s say a new car is about to come out. According to the first data, there is either a problem with some part of the vehicle every day or the chance of a crash with probability p, which means that the vehicle needs to be replaced. Also, every car that has been there for n days needs to be replaced. What is the long-term frequency of vehicle replacements?.
