Introduction
The subtopic Data & Numbers forms the quantitative backbone of the WBCS Preliminary and Main examinations. Within the broader syllabus of Quantitative Aptitude, this area tests your ability to recognise numerical patterns, apply fundamental arithmetic operations, manipulate exponents and surds, solve counting problems, handle date‑and‑calendar logic, and work with elementary set theory. In the last eight years (2016–2023), WBCS has asked 17 questions directly from this subtopic, covering a rich variety of sub‑themes: number sequences, composite numbers, exponential equations, digit counting, permutation and combination applications, perfect square constructions, calendar calculations, inequalities, and pattern matching with alphanumeric series. The 2018 paper, for instance, included a question on symbolic operations where the correct answer was "36-6+3x5÷3=74", and another on identifying a person who is neither hardworking nor ambitious.
The examiners are not looking for rote memorisation; they want you to think flexibly with numbers. Many questions are deceptively simple—like “what is the smallest composite number?”—yet a surprising number of aspirants trip because they confuse composite with prime. Others involve multi‑step reasoning, such as the 2020 problem where a cyclic sum condition forces you to deduce a specific term. The difficulty range is from straightforward (WBCS 2016, “how many days between two dates?”) to moderately tricky (WBCS 2020, exponential equation yielding three unknowns). No question requires calculus or advanced algebra; all are solvable with a solid grasp of middle‑school mathematics and careful logical application.
This chapter is designed to be your comprehensive reference. We will start from first principles—defining every jargon term before using it, building conceptual foundations, and then diving deep into the specific question types that have appeared. Along the way, we will embed every PYQ (Previous Year Question) from the 17 provided, explaining not only the correct answer but also why the distractors are plausible. By the end, you will have a framework for tackling any new Data & Numbers question that WBCS can throw at you, and you will be equipped with memory aids, pattern‑recognition techniques, and a clear sense of what might appear next.
Core Concepts & Foundations
This section establishes the building blocks. Each key term is introduced in a blockquote definition, followed by elaboration. Read these carefully; they are the language in which exam problems are written.
Natural Numbers: The set of positive counting numbers {1, 2, 3, …}. Sometimes zero is excluded in competitive exams unless stated otherwise.
Whole Numbers: Natural numbers plus zero {0, 1, 2, 3, …}. Zero is neither positive nor negative.
Integers: Whole numbers plus their negatives {…, –3, –2, –1, 0, 1, 2, 3, …}.
Rational Numbers: Numbers that can be expressed as p/q where p and q are integers and q ≠ 0. Examples: 2/3, –5, 0.75. Terminating and recurring decimals are rational.
Irrational Numbers: Numbers that cannot be expressed as p/q. Their decimal expansion is non‑terminating and non‑recurring. Examples: √2, π.
Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, … (2 is the only even prime).
Composite Number: A natural number greater than 1 that is not prime—i.e., it has at least one divisor other than 1 and itself. The smallest composite number is 4 (2 × 2). Numbers like 1 are neither prime nor composite (tested in WBCS 2020).
Place Value: The value of a digit based on its position in a number. In 473, the digit 4 stands for 400, 7 for 70, 3 for 3. This concept is crucial for digit counting problems.
Digit: A single numerical symbol (0–9). When counting how many times a digit appears in a range, we consider each occurrence.
Exponent / Power: Notation aⁿ means a multiplied by itself n times. Here a is the base, n the exponent. Laws of exponents are essential for simplifying expressions.
Factorial: Denoted n! = n × (n–1) × (n–2) × … × 1. 0! = 1 by definition. Factorials appear in permutation and combination formulas.
Permutation (nPr): The number of ways to arrange r distinct objects from a set of n distinct objects, where order matters. Formula: nPr = n! / (n–r)! . Tested in WBCS 2023.
Combination (nCr): The number of ways to choose r objects from n, where order does not matter. Formula: nCr = n! / [r!(n–r)!] . Chess tournament problems use combination.
Set: A collection of distinct objects. Union (∪) includes all elements in either set; Intersection (∩) includes elements common to both. Venn diagrams help visualise.
Leap Year: A year divisible by 4, except centuries that are not divisible by 400. Leap years have 366 days; common years have 365. The period from 17 May 1899 to 17 May 1900 spans exactly 365 days because 1900 is not a leap year (century rule).
Inequality: A statement about the relative size of two expressions. For numbers between 0 and 1, squaring makes them smaller (e.g., 0.5 > 0.25), while taking the square root makes them larger.
With these definitions in hand, we can now proceed to the deep‑dive sections, each covering a question family extracted from the PYQs.