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Quantitative Aptitude · Permutations and Combinations

Fundamental Principles of Counting for CA Foundation

Updated 1 October 2026 · Fact-checked

The fundamental principles of counting tell you how many ways a task can be done without listing them. Use the multiplication rule (m × n) when tasks happen one after another. Use the addition rule (m + n) when you pick one option from mutually exclusive choices. Ask: is it AND or OR?

Understand Fundamental Principles of Counting

Counting problems ask: in how many ways can something be done? Listing every way works for tiny cases only. The two fundamental principles let you count large cases quickly. Everything in permutations and combinations is built on them.

The multiplication rule (rule of product) applies when a job has several stages and you must complete all of them. If stage 1 can be done in m ways and, for each of those, stage 2 can be done in n ways, the whole job can be done in m × n ways. Think of the word AND. Choose a shirt AND a pant AND a pair of shoes: you multiply.

The addition rule (rule of sum) applies when a job can be done by choosing one of several alternatives, and these alternatives cannot happen together. If one option has m ways and another has n ways, and no way is common to both, the job can be done in m + n ways. Think of the word OR. Travel by bus OR by train: you add.

To decide which rule applies, picture the job. If you do everything in a sequence, multiply. If you pick just one route out of separate cases, add. Many questions need both: split into cases (add), and inside each case count the stages (multiply).

The addition rule needs the cases to be mutually exclusive. If some outcomes belong to two cases, you would count them twice. Make the cases non-overlapping first, or subtract the overlap.

Key formulas to remember

Multiplication rule (rule of product)
Total ways = m × n × p × ...
Use when stages are done one after another and every stage must be completed. Stage 2 counts are for each choice of stage 1.
Addition rule (rule of sum)
Total ways = m + n + p + ...
Use when you choose one of several mutually exclusive cases. No outcome may belong to two cases.
Words that signal the rule
AND / then / followed by → × | OR / either / alternatively → +
A guide only. Always check the actual structure of the job.
Digits with no restriction on repetition
Number of r-digit strings from n digits = n^r
Follows from the multiplication rule when repetition is allowed. If 0 cannot lead a number, the first place has one fewer choice.
Digits without repetition
n × (n − 1) × (n − 2) × ... for r places
Each place has one fewer choice than the previous one. Fill the restricted place first.

How to solve Fundamental Principles of Counting questions

Use this method for any counting question. It stops you from guessing between × and +.

  1. 1Read the question and say in one line what you are counting, such as numbers, arrangements, routes or outfits.
  2. 2Decide if the job is done in stages (all must happen) or in separate cases (only one happens).
  3. 3List the stages or cases. For stages, write one blank per stage. For cases, make sure they do not overlap.
  4. 4Fill the restricted stage first. A condition like 'first digit not zero' or 'must be even' limits that place, so count it before the others.
  5. 5Write the number of choices in each blank, adjusting for repetition allowed or not allowed.
  6. 6Multiply across stages within a case. Add the totals of different cases.
  7. 7Check the answer for sense: it should be at least as large as any single case, and not exceed the unrestricted total.
  8. 8Match your answer to the options and mark it.

Quickest way: Blank-and-restriction method

When to use it: Use for most MCQs, where numbers, codes or selections are formed under conditions. It takes under a minute.

  1. Draw blanks for each place or stage.
  2. Fill the most restricted blank first, then the rest.
  3. Multiply the choices in the blanks.
  4. If the question has 'either/or' conditions, split into cases and add the case totals. Each case total is itself a product, so the final answer is a sum of products.
  5. Work out the structure and the count first. Look at the options only afterwards, as a sanity check. Do not remove options by guessing their size, because a case-based answer is a sum of products and need not look like a simple product.
  6. If a question needs more than three cases and the total is unclear, skip it first and return at the end. Each wrong answer costs 0.25 marks.

Common mistakes in Fundamental Principles of Counting

  • Adding when the job has stages, or multiplying when the job has alternatives.

    Students match keywords instead of picturing the job.

    Fix: Ask: do I need to complete every step (multiply) or pick only one path (add)?

  • Counting the unrestricted place first and ignoring the condition.

    Students fill blanks left to right out of habit.

    Fix: Fill the restricted place first. For an even number, choose the units digit first. For no leading zero, handle the first place carefully.

  • Forgetting that 0 cannot be the first digit of a multi-digit number.

    Digits 0 to 9 are treated as equal in every place.

    Fix: Count the first place from non-zero digits only, then the other places from what remains.

  • Using n^r when repetition is not allowed, or n × (n−1) × ... when it is allowed.

    The repetition condition is skimmed over.

    Fix: Underline 'repetition allowed' or 'distinct' in the question before you start.

  • Adding overlapping cases and counting some outcomes twice.

    Cases like 'multiple of 2' and 'multiple of 5' share outcomes.

    Fix: Rebuild the cases so they cannot overlap, for example by last digit, or subtract the common outcomes.

Worked examples

Example 1

How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if repetition of digits is allowed? (a) 60 (b) 120 (c) 125 (d) 243

Show the solution
  1. The job has three stages: hundreds, tens and units. All must be filled, so multiply.
  2. Repetition is allowed, so each place has 5 choices.
  3. Total = 5 × 5 × 5 = 125.

Answer: (c) 125

Example 2

How many even 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit? (a) 20 (b) 30 (c) 24 (d) 36

Show the solution
  1. Fill the restricted units place first. Even digits available: 0, 2, 4. Split into cases by the units digit.
  2. Case 1: units digit is 0. Hundreds can be any of the remaining 4 digits (1, 2, 3, 4). Tens can be any of the remaining 3 digits. Count = 4 × 3 = 12.
  3. Case 2: units digit is 2 or 4 (2 choices). Hundreds cannot be 0 and cannot be the digit used in units, so 3 choices. Tens can be any of the remaining 3 digits, including 0. Count = 2 × 3 × 3 = 18.
  4. The cases are mutually exclusive, so add: 12 + 18 = 30.

Answer: (b) 30

Example 3

A person can go from city A to city B by 3 buses or 2 trains. For the return from B to A, he can use 4 buses or 1 flight. If he must go by bus or train and return by bus or flight, how many round-trip choices does he have? (a) 12 (b) 20 (c) 10 (d) 25

Show the solution
  1. Outward journey: bus OR train, so add: 3 + 2 = 5 ways.
  2. Return journey: bus OR flight, so add: 4 + 1 = 5 ways.
  3. The round trip needs the outward journey AND the return journey, so multiply: 5 × 5 = 25.

Answer: (d) 25

Exam tips

  • Look for the words 'and' and 'or' in the question, but confirm by picturing the job, since the wording can mislead.
  • In number-formation questions, always check the leading zero and the even/odd condition before you multiply.
  • Work out the structure of the job and the count first. Use the options only as a sanity check afterwards, and do not remove options by estimating their size.
  • Mixed questions need both rules. Split into cases, multiply inside each case, then add the totals.
  • If a question needs long case-work and time is short, skip it and return later, since a wrong answer costs 0.25 marks.

Practice questions from Permutations and Combinations

Fundamental Principles of Counting: frequently asked questions

What is the difference between the multiplication rule and the addition rule?

The multiplication rule applies when a job is done in stages, and every stage must be completed. You multiply the number of ways. The addition rule applies when you pick one of several separate options. You add the number of ways.

How do I know whether to multiply or add in a counting problem?

Picture the job. If you must do step 1 and step 2 and step 3, multiply. If you do step 1 or step 2 but never both, add. Words like 'and' and 'or' help, but check the actual structure.

Do I need the fundamental principle if I know nPr and nCr?

Yes. The formulas for permutations and combinations come from the multiplication rule. Many exam questions with restrictions or cases are easier to solve with the principles directly.

Why must the cases be mutually exclusive for the addition rule?

If one outcome belongs to two cases, adding the case totals counts it twice. Rework the cases so they do not overlap, or subtract the common outcomes.