Permutation, Combination and Probability Formula Guide: nPr, nCr, Circular Arrangements and Probability Rules for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
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Permutation, combination and probability are the “counting” chapters of SSC maths, and all questions depend on a handful of formulas. This permutation, combination and probability formula guide for SSC gives the formulas in tables with the standard data of coins, dice and cards, along with solved examples, practice questions and FAQs.
Basic Counting Principles
| Principle | Meaning |
|---|---|
| Fundamental principle of multiplication | If one task can be done in m ways and another in n ways, both can be done in m × n ways |
| Principle of addition | If a task can be done in m ways or in n ways (not both), the total is m + n ways |
| Factorial | n! = n × (n − 1) × … × 2 × 1; 0! = 1; 5! = 120; 6! = 720 |
Permutations and Combinations
| Item | Formula |
|---|---|
| Permutation (arrangement) of n things taken r at a time | nPr = n!/(n − r)! |
| Combination (selection) of n things taken r at a time | nCr = n!/[r!(n − r)!] |
| Relation | nPr = nCr × r! |
| Symmetry | nCr = nCn−r |
| Pascal’s rule | nCr + nCr−1 = n+1Cr |
| nC0 and nCn | Both equal 1 |
| Arranging n different things in a row | n! |
| Arranging n things with p alike of one kind, q alike of another | n!/(p! q!) |
| Circular arrangement of n different things | (n − 1)! |
| Circular arrangement with clockwise and anticlockwise treated as the same (necklace) | (n − 1)!/2 |
| Number of ways to choose any number of things from n | 2n − 1 (excluding the case of choosing none) |
| Handshakes among n persons | nC2 = n(n − 1)/2 |
| Diagonals of a polygon of n sides | nC2 − n = n(n − 3)/2 |
| Number of triangles from n points (no three collinear) | nC3 |
Words and Arrangements: Tricks
- If some things must always be together, treat them as one unit and then arrange the units and the things inside.
- If some things must never be together: total arrangements minus the arrangements when they are together.
- Vowels and consonants in the alternate places: arrange each group in its places separately and multiply.
- In a word with repeated letters, divide by the factorial of the repeats.
- Rank of a word in the dictionary: count the arrangements of the words that come before it.
Probability Formulas
| Item | Formula |
|---|---|
| Probability of an event | P(E) = number of favourable outcomes / total number of outcomes |
| Range | 0 ≤ P(E) ≤ 1; a certain event has probability 1 and an impossible event 0 |
| Complement | P(not E) = 1 − P(E) |
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) |
| Mutually exclusive events | P(A or B) = P(A) + P(B) |
| Independent events | P(A and B) = P(A) × P(B) |
| Conditional probability | P(A given B) = P(A and B)/P(B) |
| Odds in favour | Favourable : unfavourable |
Standard Data for Probability
| Experiment | Total Outcomes |
|---|---|
| 1 coin | 2 (H, T) |
| n coins | 2n |
| 1 die | 6 |
| 2 dice | 36 |
| n dice | 6n |
| Pack of cards | 52 cards: 4 suits (hearts and diamonds red; spades and clubs black) of 13 each |
| Face cards (kings, queens, jacks) | 12 cards |
| Honour cards | 16 (A, K, Q, J of each suit) or 20 (including the 10s), as defined in the question |
| Sum of two dice | Sum 7 has 6 outcomes; sum 2 and sum 12 have 1 each |
Solved Examples
Example 1. Find 5P2.
5!/3! = 20.
Example 2. Find 6C2.
(6 × 5)/2 = 15.
Example 3. In how many ways can the letters of the word MISSISSIPPI be arranged?
11!/(4! × 4! × 2!) = 39,916,800/1,152 = 34,650.
Example 4. In how many ways can 5 persons sit around a circular table?
(5 − 1)! = 24.
Example 5. Find the probability of getting a sum of 7 when two dice are thrown.
6/36 = 1/6.
Example 6. Find the probability of drawing a king from a pack of 52 cards.
4/52 = 1/13.
Example 7. Find the probability of getting at least two heads when three coins are tossed.
The favourable outcomes: HHH, HHT, HTH, THH = 4; the probability = 4/8 = 1/2.
Quick Tips
- Use permutations when the order matters and combinations when it does not.
- Remember the circular arrangement (n − 1)!.
- Use the complement for the “at least one” questions.
- Remember that two dice have 36 outcomes and three coins have 8.
Quick Revision Points
- nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!].
- Circular arrangements: (n − 1)!.
- P(E) = favourable/total.
- P(not E) = 1 − P(E).
- Two dice: 36 outcomes; a pack has 52 cards.
Practice Questions With Answers
- Find 7C3.
Answer: 35 - Find 6P3.
Answer: 120 - In how many ways can the letters of the word BANANA be arranged?
Answer: 60 - In how many ways can 6 persons sit around a round table?
Answer: 120 - Find the probability of a head in a single toss of a coin.
Answer: 1/2 - Find the probability of getting a prime number when a die is thrown.
Answer: 1/2 - Find the probability of getting a red card from a pack.
Answer: 1/2 - How many handshakes are there among 12 persons?
Answer: 66 - Find the number of diagonals of a hexagon.
Answer: 9 - Find the probability of getting a sum of 12 with two dice.
Answer: 1/36
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Formula Guide notes page, read the previous topic Trains, Boats and Streams Formula Guide and the next topic Fractions, Percentages and Tables Quick Reference.
Frequently Asked Questions
What is the difference between permutation and combination?
Permutation counts arrangements where the order matters; combination counts selections where it does not.
What is the formula for nPr?
n!/(n − r)!.
What is the formula for nCr?
n!/[r!(n − r)!].
What is a factorial?
The product of all positive integers up to n.
How many ways can n persons sit in a circle?
(n − 1)!.
What is the probability of an event?
The number of favourable outcomes divided by the total number of outcomes.
What is the probability of an impossible event?
Zero.
How many cards are in a pack?
52.
How many outcomes do two dice have?
36.
Is this topic asked in SSC?
Yes. Counting and probability are asked in SSC CGL.
What is the complement rule?
P(not E) = 1 − P(E).
What is the value of 0!?
1.
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Trains, Boats and Streams Formula Guide: Speed, Crossing Time and Upstream Downstream Formulas for SSC Maths
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Fractions, Percentages and Multiplication Tables Quick Reference: Fraction to Percentage Table and Calculation Tricks for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.