Combination Calculator

Combination Formula
nCr = n ! /r! ( n - r )!

Example 1. A teacher must select 3 students from a list of 6 students. How many distinct groups of 3 students are possible?

Example 2. A committee consisting of 5 teachers will be chosen from a staff of 25 teachers. Find the number of possible combination of 5 teacher committees.

n = Given, total number of distinct element to choose from.

n ! =

r = Given, taking unique combination of element at a time meaning no repetition of same element or double counting.

r ! =

Click this button → to solve for combination

(n- r)! =

Answer: n C r = Possible combination of unique element meaning no repetition or double counting. Arrangement reversal or order reversal are not counted.


The result of combination is your all possible solution set and it is use to calculate the p-value or probability value.
For example : 8 C 4 = 70 possible combination. The probability of getting 1 correct instance out of 70 possible combination is 1/70 or 0.0143 or 1.43% . In inferential or hypothesis testing industry practice, if the probability value is =< 5% (p-value =<5%), the conclusion is the intervention program is working. For example, if the reported p-value of the study is 1.43% then the conclusion is the intervention program is working because there is only 1.43% probability that it will happen by random or by chance.

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Important thing to remember in combination. No double counting, no repetition, no arrangement reversal or no order reversal is allowed as correct answer. In the above image, You will notice from count # 1 up to the last count # 20, There is no double counting of color, each color is unique. No repetition of color is allowed in combination.

For letters Given data = { A, B, C, D } Solve for all possible combination using 2 letters at a time. Counting all the possible combination. Combination number 1 = AB here if you reverse the order of B, BA is not a valid answer, so you will not count it because the ingredient or component are still the same AB = BA. Combination number 2 = AC. Solution = { AB, AC, AD, BC, BD, CD }. If you count the total combination the answer is six possible combination. You will get the same answer if you use the formula. You see mathematician uses formula to speed up the listing process and save time especially if you are solving 26 possible combination of two letters in English alphabet. Try listing and then counting all the possible two letters combination in English alphabet. If you really try it, time your starting time and your finish time and record how long it will take you to arrive at the correct answer of 325 using the formula in less than 10 seconds.

How about listing and then counting all the possible three letters combination , how long will it take you to do it manually to arrive at the correct answer of 2,600 using the formula in less than 10 seconds? We should thank the mathematicians for discovering the general rule , also called algorithm or formula using inductive reasoning process.

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Machine learning simple explanation

We can see how a machine learning works by using how a mathematician solve a problem. Mathematician think in general or global perspective they say let variable letter n to represent any possible real number of given population under research. Next let variable letter r to represent any possible grouping or combination at a snap shot of time. Next they use the tool called factorial designated by symbol exclamation mark !. That's all the abstract concept they use to solve the problem of spending too much time doing manual listing and counting process in order to answer the question how many possible combination. Imagine how many unanswered question out there because we don't have the understanding and tool to solve it. But with Artificial Intelligence , we will be able to answer some of many unanswered questions out there.

We don't question the combination formula if the answer is valid or correct because we are convince it is the correct answer and we can prove it or show it.

Artificial intelligence specifically machine learning are using many possible combination of inputs by approximation to compute the Expected most likelihood of correct answer. The correct answers or ground truths or first principles are called the training data.

Human should pay close attention in understanding, certifying, and governing the training data or ground truth or first principles used as training data. Once humans are convince that the training data are invariant or consistent and stable at known date range of training. Then the expected outcome is explainable and traceable using search algorithm if needed for audit purposes.

Machine learning will do the iteration or trial and error or approximation using the correct answer or first principle from the training data sets until the human subject matter expert will decide to stop the approximation because the likelihood of getting correct answer is 95% at a time or 99% at a time then document the reason for accepting the risk and the mitigation of risk of failure.

Human subject matter engineer or expert or scientist use the calculated answer and add safety or insurance policy to compensate for the error. As we get better in explaining how the correct answer was derived the use of A.I. will grow exponentially and more things that seems impossible will become possible.

Performance Objective:

After this tutorial the students are expected to solve listing and counting problems and represent counting principle algebraically including factorial notation.

There are 6 books on a reading list. Students must read 3 of the books on the list. In how many different ways can a student select 3 books?

The above problem is a combination problem. To learn how to count the possible combination . You must remember you only count the distinct combination, meaning no repetition or double counting of element.

For example the combination ( book1, book2, book3 ) is one (1) count

the combination (book2, book1,book3) is not distinct combination from the above combination because you are double counting the same element at that instance, so it is not counted. Furthermore the combination (book3,book1,book2) is not different or distinct from the above combination so it is also not counted.

Now let us do the manual counting of the possible combination to solve the above problem.
REMEMBER ONLY COUNT DISTINCT COMBINATION, NO DOUBLE COUNTING

Using the above table there are only 20 possible distinct combination or 20 possible ways to select 3 books out of the six books.

Using Excel spreadsheet, you solve the problem using the formula COMBIN. You will get the same result, which is 20.

Easy Question - A teacher must select 2 students from a list of 4 students. How many distinct groups of 2 students are possible?

Difficult Question - A committee consisting of 5 teachers will be chosen from a staff of 25 teachers. Find the number of possible combination of 5 teacher committees.

Next lesson is PERMUTATION. Click here.

Can you explain the difference between combination and permutation. Give example to support your answer.

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How can IN-V-BAT-AI be used in classrooms ?

IN-V-BAT-AI is a valuable classroom tool that enhances both teaching and learning experiences. Here are some ways it can be utilized:

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Personal Augmented Intelligence (AI) Explanation

🧠 Augmented Intelligence vs Artificial Intelligence

Understanding the difference between collaboration and automation

🔍 Messaging Contrast

Augmented Intelligence is like a co-pilot: it accelerates problem-solving through trusted automation and decision-making, helping you recall, analyze, and decide — but it never flies solo.

Artificial Intelligence is more like an autopilot: designed to take over the controls entirely, often without asking.

💡 Why It Matters for IN-V-BAT-AI

IN-V-BAT-AI is a textbook example of Augmented Intelligence. It empowers learners with one-click recall, traceable results, and emotionally resonant memory tools. Our “Never Forget” promise isn't about replacing human memory — it's about enhancing it.

Note: This is not real data — it is synthetic data generated using Co-Pilot to compare and contrast IN-V-BAT-AI with leading EdTech platforms.

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