Permutation, Combination and Probability for CSAT

Permutation, combination and probability look difficult at first, but in the UPSC CSAT paper, the questions are at the level of Class 10 and are usually based on simple counting and everyday situations: arranging people, forming committees, tossing coins, throwing dice or drawing cards. Once you understand the logic of "counting the ways", you can solve these questions quickly and accurately. This note explains the basic counting principles, permutations, combinations and probability in simple language, with formulas, shortcuts and many solved examples, so that you can handle these questions in the CSAT and other competitive examinations.

Fundamental Counting Principles

The base of the chapter is two simple rules. The multiplication principle says that if one task can be done in m ways and, after that, another task can be done in n ways, then the two tasks together can be done in m × n ways. For example, a person has 4 shirts and 3 trousers, so the number of different outfits is 4 × 3 = 12. The addition principle says that if one task can be done in m ways or another task can be done in n ways (and not both at the same time), the total number of ways is m + n. For example, if there are 3 bus routes and 2 train routes to a city, a traveller can go in 3 + 2 = 5 ways.

Factorial

The factorial of a positive integer n, written n!, is the product of all positive integers from 1 to n: n! = n × (n − 1) × (n − 2) × ... × 2 × 1. So 5! = 120, 4! = 24, 3! = 6, and by definition 0! = 1. Factorials count the number of ways to arrange n different things in a row.

Permutations: When Order Matters

A permutation is an arrangement in which the order is important. The number of permutations of n different things taken r at a time is nPr = n!/(n − r)!. Examples:

• In how many ways can 3 persons sit on 5 chairs? 5P3 = 5!/2! = 60.
• How many 3-digit numbers can be formed from the digits 1 to 5 without repetition? 5P3 = 60. If repetition is allowed, there are 5 × 5 × 5 = 125.
• In how many ways can the letters of the word CAT be arranged? 3! = 6.

Permutations with repeated items: If there are n items in which p are alike of one kind, q are alike of another kind and so on, the number of arrangements is n!/(p! × q! × ...). Example: the word LEVEL has 5 letters, with L twice and E twice. The number of arrangements is 5!/(2! × 2!) = 120/4 = 30.

Circular permutations: When n different people sit around a round table, the arrangements are (n − 1)!, because the rotation of the whole group gives the same arrangement. For 5 people, it is 4! = 24.

Combinations: When Order Does Not Matter

A combination is a selection in which the order is not important. The number of combinations of n different things taken r at a time is nCr = n!/(r! × (n − r)!). Also, nCr = nC(n − r), and nPr = nCr × r!. Examples:

• In how many ways can a committee of 3 be chosen from 6 people? 6C3 = (6 × 5 × 4)/(3 × 2 × 1) = 20.
• From 5 men and 4 women, choose 2 men and 2 women: 5C2 × 4C2 = 10 × 6 = 60.
• In a party of 10 people, if everyone shakes hands with everyone else once, the number of handshakes is 10C2 = 45. (General formula: n(n − 1)/2.)
• The number of diagonals in a polygon of n sides is nC2 − n = n(n − 3)/2; for a hexagon, this is 6 × 3/2 = 9.

Permutation vs Combination: A Quick Test

A good tip is to ask: "Will changing the order change the answer?" If yes, use permutation; if no, use combination.

Probability: Measuring Chance

Probability is a number between 0 and 1 that measures the chance of an event. The formula is P(E) = Number of favourable outcomes / Total number of outcomes, when all outcomes are equally likely. A probability of 0 means an impossible event, and 1 means a certain event. The sum of the probabilities of an event and its complement is 1: P(not E) = 1 − P(E).

Basic Examples

• Coin: When a coin is tossed, P(head) = 1/2. When 3 coins are tossed, there are 8 outcomes, and the probability of exactly 2 heads is 3/8 (HHT, HTH and THH).
• Die: When a die is thrown, P(a prime number) = 3/6 = 1/2 (2, 3 and 5). P(an even number) = 3/6 = 1/2.
• Two dice: There are 36 outcomes. P(sum 7) = 6/36 = 1/6 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). P(sum 8) = 5/36.
• Cards: A pack has 52 cards, with 4 aces, 13 cards of each suit and 26 red cards. P(an ace) = 4/52 = 1/13. P(a heart) = 13/52 = 1/4.
• Balls from a bag: A bag has 4 red and 6 blue balls. If two balls are drawn at random, P(both red) = 4C2/10C2 = 6/45 = 2/15.

Rules of Probability

Two events are mutually exclusive if they cannot happen together (a coin cannot show both head and tail in one toss). They are independent if the result of one does not affect the other (two separate coin tosses). Conditional probability, P(A given B) = P(A and B)/P(B), is used when the second event depends on the first; for example, if two cards are drawn without replacement, the chance of the second being an ace depends on what the first card was.

More Solved Examples

Example 1. Two dice are thrown. What is the probability that the product is even? The product is odd only when both numbers are odd: 3 × 3 = 9 outcomes out of 36, so P(odd) = 1/4, and P(even) = 1 − 1/4 = 3/4.

Example 2. From a group of 6 boys and 4 girls, a team of 4 is selected. What is the probability that it has exactly 2 girls? Total ways = 10C4 = 210. Favourable ways = 4C2 × 6C2 = 6 × 15 = 90. So the probability is 90/210 = 3/7.

Example 3. In how many ways can 5 people be seated in a row if two particular people must always sit together? Treat the two as one unit, so there are 4 units: 4! = 24 ways, and the two can swap places among themselves in 2! = 2 ways. The total is 24 × 2 = 48.

Tips for the CSAT

Read the question and decide if order matters. Write the formula before putting the numbers. For "at least one" questions, use the complement. For probability with cards, dice and coins, list the total outcomes first (52, 36, 2^n). Use cancellation to simplify nCr before multiplying: 10C3 = (10 × 9 × 8)/(3 × 2 × 1) = 120. If options are far apart, estimate the answer. Check that a probability is always between 0 and 1. Practise 10 questions a day for a week.

Common Mistakes

Students often use permutation when the order does not matter, forget to divide by the factorials of repeated items, count the same case twice, mix up "and" (multiplication) with "or" (addition), and forget the complement method. Another mistake is to treat events as independent when drawing without replacement. Take a moment to think about whether the objects are distinct, whether repetition is allowed and whether the order matters.

Quick Revision

• nPr = n!/(n − r)!; nCr = n!/(r!(n − r)!); circular arrangement of n people: (n − 1)!.
• Repeated items: n!/(p! q! ...); handshakes: nC2.
• P(E) = favourable/total; P(not E) = 1 − P(E); "at least one" = 1 − P(none).
• Dice: 36 outcomes; cards: 52; coins: 2^n.

These topics reward logical thinking and practice. Learn the formulas, solve the standard examples and try a few new problems every day, and you will turn this part of the CSAT into reliable marks.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Decision Making and Interpersonal Skills for CSAT
• Number System, HCF and LCM Notes for UPSC CSAT with Solved
• Basic Geometry and Mensuration Notes for UPSC CSAT with Solved
• All CSAT notes

Frequently Asked Questions (FAQ)

What is a permutation?

A permutation is an arrangement of objects in a particular order. The number of permutations of n things taken r at a time is nPr = n!/(n − r)!.

What is a combination?

A combination is a selection of objects in which the order does not matter. The number of combinations is nCr = n!/(r!(n − r)!).

What is the difference between permutation and combination?

In permutations, the order of arrangement matters, while in combinations, only the selection matters and order does not.

What is a factorial?

The factorial of n, written n!, is the product of all positive integers up to n. For example, 5! = 120.

How many ways can n people sit around a circular table?

n different people can be seated around a circular table in (n − 1)! ways.

What is probability?

Probability is a measure of the chance that an event will occur, calculated as the number of favourable outcomes divided by the total number of outcomes.

What is the probability of getting a head in a coin toss?

The probability of getting a head in one toss of a fair coin is 1/2.

How many outcomes are there when two dice are thrown?

There are 36 outcomes when two dice are thrown.

What is the probability of drawing an ace from a deck of 52 cards?

There are 4 aces in 52 cards, so the probability is 4/52 = 1/13.

What are mutually exclusive events?

Mutually exclusive events are events that cannot occur at the same time, such as getting a head and a tail in the same toss.

How do I solve "at least one" probability questions?

Find the probability of the event that none occurs and subtract it from 1.

How many handshakes occur among 10 people if each shakes hands once with everyone?

The number of handshakes is 10C2 = 45.