Mental Ability: Probability, Permutation, and Combination — Formulas and Solved PSC Problems

Permutation, Combination, and Probability questions appear in almost every Kerala PSC graduate-level exam (1-3 questions). This note covers all formulas, concepts, and 10 fully solved problems in the PSC pattern.

Fundamental Counting Principle

Principle	Rule
Multiplication Principle	If task A can be done in m ways AND task B in n ways, together they can be done in m x n ways
Addition Principle	If task A can be done in m ways OR task B in n ways (mutually exclusive), total = m + n ways

Factorial

Symbol	Meaning	Example
n!	n x (n-1) x (n-2) x … x 2 x 1	5! = 5 x 4 x 3 x 2 x 1 = 120
0!	1 (by definition)	—
1!	1	—

Value	Result
2!	2
3!	6
4!	24
5!	120
6!	720
7!	5,040
8!	40,320
9!	3,62,880
10!	36,28,800

Permutation (nPr) — Order MATTERS

Formula	nPr = n! / (n-r)!
Meaning	Number of ways to ARRANGE r items from n items
When to use	When ORDER matters (first, second, third…)
Example	Arranging 3 books from 5 on a shelf

Scenario	Formula
Arrange r from n	nPr = n!/(n-r)!
Arrange all n items	n!
Circular arrangement of n	(n-1)!
Arrange with repetition (p items same, q items same)	n! / (p! x q!)
Arrange r from n with repetition allowed	n^r

Combination (nCr) — Order does NOT matter

Formula	nCr = n! / [r! x (n-r)!]
Meaning	Number of ways to SELECT r items from n items
When to use	When ORDER does not matter (just choosing)
Example	Selecting 3 members from 5 for a committee

Key Properties	Formula
nC0 = nCn	1
nC1	n
nCr = nC(n-r)	Symmetry (10C3 = 10C7)
nCr + nC(r-1)	(n+1)Cr (Pascal’s rule)

Common Values (Memorize for Speed)

Expression	Value
10C2	45
10C3	120
12C2	66
52C1	52
6C2	15
8C3	56
5C2	10
5C3	10

Probability Basics

Concept	Formula
Probability of event A	P(A) = Favorable outcomes / Total outcomes
Range	0 to 1 (0 = impossible, 1 = certain)
Complement	P(not A) = 1 - P(A)
Sure event	P = 1
Impossible event	P = 0

Types of Events

Type	Definition	P(A or B)
Mutually Exclusive	Cannot happen together (A and B = empty)	P(A) + P(B)
Non-mutually Exclusive	Can happen together	P(A) + P(B) - P(A and B)
Independent	Occurrence of one does not affect other	P(A and B) = P(A) x P(B)
Dependent	Occurrence of one affects other	P(A and B) = P(A) x P(B given A)

Key Probability Formulas

Scenario	Formula
P(A or B) — general	P(A) + P(B) - P(A and B)
P(A or B) — mutually exclusive	P(A) + P(B)
P(A and B) — independent	P(A) x P(B)
P(at least one)	1 - P(none)
Odds in favor	Favorable : Unfavorable
Odds against	Unfavorable : Favorable

Solved PSC-Style Problems

Problem 1: Basic Permutation

Q: In how many ways can 5 people be seated in a row?

Step	Calculation
This is arrangement of all 5	5! = 120
Answer	120

Problem 2: Combination (Committee)

Q: From 8 men and 5 women, a committee of 5 is to be formed with 3 men and 2 women. How many ways?

Step	Calculation
Select 3 men from 8	8C3 = 56
Select 2 women from 5	5C2 = 10
Total (multiply)	56 x 10 = 560
Answer	560

Problem 3: Word Arrangement

Q: How many words can be formed from the letters of “KERALA”?

Step	Calculation
Total letters	6
Repeated: A appears 2 times	—
Formula	6! / 2! = 720/2 = 360
Answer	360

Problem 4: Circular Arrangement

Q: In how many ways can 6 people sit around a round table?

Step	Calculation
Circular arrangement formula	(n-1)!
(6-1)! = 5!	120
Answer	120

Problem 5: Probability (Dice)

Q: Two dice are thrown. What is the probability of getting a sum of 7?

Step	Calculation
Total outcomes	6 x 6 = 36
Favorable: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)	6 outcomes
P = 6/36	1/6
Answer	1/6

Problem 6: Probability (Cards)

Q: A card is drawn from a pack of 52. What is the probability of getting a King or a Heart?

Step	Calculation
P(King)	4/52
P(Heart)	13/52
P(King AND Heart) = King of Hearts	1/52
P(King OR Heart) = 4/52 + 13/52 - 1/52	16/52 = 4/13
Answer	4/13

Problem 7: Probability (At least one)

Q: A coin is tossed 3 times. What is the probability of getting at least one head?

Step	Calculation
P(no head) = P(all tails)	(1/2)^3 = 1/8
P(at least one head) = 1 - P(no head)	1 - 1/8 = 7/8
Answer	7/8

Problem 8: Combination (Selection)

Q: In how many ways can 4 letters be selected from the word “PROBLEM”?

Step	Calculation
Total distinct letters in PROBLEM	7
Select 4 from 7	7C4 = 7C3 = 35
Answer	35

Problem 9: Probability (Balls in Bag)

Q: A bag contains 5 red and 3 blue balls. Two balls are drawn at random. What is the probability that both are red?

Step	Calculation
Total balls	8
Total ways to draw 2	8C2 = 28
Ways to draw 2 red from 5	5C2 = 10
P(both red) = 10/28	5/14
Answer	5/14

Problem 10: Permutation (Number Formation)

Q: How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition?

Step	Calculation
Hundreds place: 5 choices	5
Tens place: 4 remaining	4
Units place: 3 remaining	3
Total = 5 x 4 x 3	60
Answer	60 (same as 5P3)

Quick Decision: Permutation or Combination?

Keyword in Question	Use	Why
Arrange, seat, order, rank, word formation	Permutation	Order matters
Select, choose, committee, group, team	Combination	Order does not matter
”In how many ways can X be arranged”	Permutation	Arrangement
”In how many ways can X be selected”	Combination	Selection

PSC Exam Shortcuts

Shortcut	Application
nC2 = n(n-1)/2	Quick calculation for “choose 2”
Handshakes among n people	nC2 = n(n-1)/2
Diagonals of n-sided polygon	nC2 - n = n(n-3)/2
Matches in tournament (each plays each)	nC2
P(at least 1) = 1 - P(none)	Fastest method for “at least” questions

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Memory Aid: Permutation = Position matters (arrangement). Combination = Choice only (selection). “The COMMITTEE doesn’t care who sits where” = Combination.