CMA Foundation · Fundamentals of Business Mathematics and Statistics
Permutation and Combinations for CMA Foundation Paper 3
Permutation counts the ways to arrange objects where order matters. Combination counts the ways to select objects where order does not matter. To solve a question, decide whether order matters, then use nPr = n! ÷ (n − r)! or nCr = n! ÷ [r! (n − r)!], or the multiplication and addition principles.
What this chapter covers
This chapter teaches you to count without listing. You learn how many ways a task can be done, how many arrangements exist, and how many selections can be made. The tools are small: the two counting principles, factorials, nPr and nCr.
The chapter is built in layers. First you learn the multiplication and addition principles. Then factorials give you a short way to write products. Permutations come next, including cases with repeated objects and restrictions. Combinations follow, and the last topic mixes selection and grouping in committee-type problems.
In Paper 3 this chapter links closely with Probability, where you count favourable and total outcomes using nCr. It also supports the Binomial theorem idea of nCr as coefficients. The calculations are short, so it is a good place to score quickly in a one-hour objective paper.
Questions here are usually short, formula-based MCQs with clean numbers, so you can solve many in under a minute each. Since there is no negative marking, you can attempt every one. The same counting skills are reused in Probability, so a firm base here pays off in more than one chapter. The effort is mostly in learning to read the question correctly, not in heavy calculation.
Permutation and Combinations: topics in the order to study them
- 1Fundamental Principles of CountingEvery later formula rests on the multiplication and addition principles, so learn when to multiply and when to add first.
- 2Factorial Notation and Its PropertiesnPr and nCr are written with factorials, so you need to simplify them fast before using formulas.
- 3Permutations of Distinct ObjectsThis is the first direct use of factorials: arranging objects where order matters.
- 4Permutations with Repetition and RestrictionsIt extends arrangements to repeated letters and conditions like 'together' or 'never together', which need the basic case to be solid.
- 5Combinations and the Formula nCrOnce you know permutations, you see combinations as permutations divided by r!, because order is ignored.
- 6Applications of Combinations: Selection and GroupingIt comes last because it mixes all earlier ideas in committee, team and selection problems.
How to prepare Permutation and Combinations
Spend your time on understanding the type of question, then practise until the choice of method is automatic.
- Learn the two counting principles with everyday examples: 'and' means multiply, 'or' means add.
- Practise factorial simplification until you can cancel terms like 8! ÷ 6! = 8 × 7 = 56 without writing everything out.
- Memorise nPr = n! ÷ (n − r)! and nCr = n! ÷ [r! (n − r)!], and the facts nC0 = nCn = 1 and nCr = nC(n − r).
- For every problem, ask one question first: does order matter? Write 'arrange' or 'select' next to the question before computing.
- Solve restriction problems by treating a 'together' group as one object, then arranging the group inside itself.
- Do timed MCQ sets of 15 to 20 questions, and note down each wrong answer with the reason.
- In the last week, revise only the formulas and your error notebook, then redo mixed questions.
Common mistakes in Permutation and Combinations
Using a permutation when the question asks for a selection, or the reverse.
Fix: Underline words like arrange, order, rank (permutation) and select, choose, committee (combination) before you compute.
Adding where you should multiply, or multiplying where you should add.
Fix: Ask if the choices happen in sequence (multiply) or are alternatives (add).
Writing 0! = 0.
Fix: Remember 0! = 1, which makes nCn = 1 and nC0 = 1 work.
Forgetting to divide by the factorials of repeated letters.
Fix: List the repeated letters first, for example in a word, and divide n! by each repeat's factorial.
Forgetting the internal arrangements of a group that must stay together.
Fix: Always multiply by the factorial of the number of items inside the group.
Computing full factorials instead of cancelling.
Fix: Cancel common terms first, for example 10C2 = (10 × 9) ÷ 2 = 45.
Last-day revision: Permutation and Combinations
- Multiplication principle: if tasks happen one after another, multiply the ways.
- Addition principle: if tasks are alternatives and cannot happen together, add the ways.
- n! = n × (n − 1) × ... × 2 × 1, and 0! = 1.
- nPr = n! ÷ (n − r)!, and nPn = n!.
- nCr = n! ÷ [r! (n − r)!], and nPr = nCr × r!.
- nCr = nC(n − r), so use the smaller r to calculate faster.
- nC0 = nCn = 1 and nC1 = n.
- Arrangements of n objects where p are alike of one kind and q alike of another: n! ÷ (p! q!).
- Items that must stay together: treat them as one object, then multiply by their internal arrangements.
- Never together = total arrangements − arrangements with them together.
- Circular arrangement of n distinct objects: (n − 1)!.
- If order matters it is a permutation; if it does not, it is a combination.
Permutation and Combinations practice questions
- A company in Pune must choose one person for a task, either from 6 accountants or from 5 engineers, and no person belongs to both groups. Se…
- A committee of 3 members is to be chosen from 8 officers of a firm in Pune. In how many ways can the committee be formed?
- A Chennai firm must assign 4 different tasks to 4 different employees, one task per employee. In how many different ways can the assignment …
- A Mumbai firm assigns 4-letter codes to its cost centres using the letters of the word 'FUND' with no letter repeated. How many different co…
- A committee of 3 members is to be chosen from 8 employees of Sharma Traders. In how many ways can the committee be chosen?
- If (n+1)! = 30 × (n-1)!, what is n?
- A traveller from Delhi to Kolkata can go by 3 train routes or by 2 flight routes, and for the return journey, the traveller can use any of 4…
- How many different 4-digit numbers, with no digit repeated, can be formed using the digits 1, 2, 3, 4, 5, 6 so that the number is even?
Permutation and Combinations in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Permutation and Combinations: frequently asked questions
How do I know whether to use permutation or combination?
Check if changing the order gives a different result. Arranging people in a queue gives different results, so it is a permutation. Choosing a team of people does not, so it is a combination.
Is this chapter in Paper 3 only?
Yes, it belongs to Paper 3, Fundamentals of Business Mathematics and Statistics. Its counting methods are also useful for the probability questions in the same paper.
Which formulas must I remember by heart?
Learn n!, nPr, nCr, nCr = nC(n − r), and the formula for arrangements with repeated objects. Most MCQs can be solved with these.
How long should I spend on a question in the exam?
The paper has 50 questions in one hour, so you have about a minute each on average. Counting questions are usually short, so try to finish them quickly and save time for harder ones. There is no negative marking, so attempt every question.