Counting, Selections and Arrangements Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Counting, Selections and Arrangements notes for XAT: 14 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Counting, Selections and Arrangements

Chapter Roadmap

1. Fundamental Counting Principle

The bedrock of all counting. Mastering the 'AND' and 'OR' rules.

2. Permutations: Art of Arrangement

When order matters. Linear, circular, and identical items.

3. Combinations: Art of Selection

When order does not matter. Grouping and powerful properties.

4. Grouping and Distribution

Dividing items. Identical vs distinct groups.

5. Advanced Constraints

Complex conditions, overlapping categories, and geometric arrangements.

The Core Intuition: And vs Or

The Core Intuition: AND vs OR

Multiplication (AND)

Independent stages connected by AND.

Example: Choose a shirt AND a pant.

Action: Multiply.

Addition (OR)

Mutually exclusive alternatives connected by OR.

Example: Travel by train OR by bus.

Action: Add.

First-Timer Tip: Frame the problem in plain English. If stages happen to complete one goal, multiply. If they are separate ways to achieve the same goal, add.

Permutations vs Combinations: The Order Test

Permutations vs Combinations

Permutation (Arrangement)

Question: Does the sequence or order matter?

Example: Forming a 3-digit number (123 $\neq =$ 321).

Formula: $^{n} P_{r} = \frac{n !}{( n - r )!}$

Combination (Selection)

Question: Does order not matter? Just a group?

Example: Choosing a 3-member committee (A,B,C = C,B,A).

Formula: $^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$

The Golden Rule: If swapping two items creates a "new" outcome, it is a permutation. If swapping leaves it unchanged, it is a combination.

The Essential Formulas and Properties

Essential Formulas and Properties

Core Formulas

^{n} P_{r} = \frac{n !}{( n - r )!}

^{n} C_{r} = \frac{n !}{r ! ( n - r )!}

^{n} P_{r} =^{n} C_{r} \times r!

Powerful Properties

Symmetry:

^{n} C_{r} =^{n} C_{n - r}

Choosing

r

to keep is same as

n - r

to leave.

Pascal's Identity:

^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}

Crucial for summing series.

Sum of Combinations:

^{n} C_{0} +^{n} C_{1} + ... +^{n} C_{n} = 2^{n}

Total subsets of a set with

n

elements.

Counting, Selections and Arrangements: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

What is the fundamental test to decide whether a counting problem requires permutations or combinations?

A.
Whether the items are distinct or identical.
B.
Whether the order of the selected items matters.
C.
Whether the selections are made with or without replacement.
D.
Whether the problem involves addition or multiplication.

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition question testing the core distinction between permutations and combinations.

Step 1: Permutations deal with arrangements where the sequence or position of items is important.

Step 2: Combinations deal with selections where only the identity of the chosen items matters, not their sequence.

Step 3: Therefore, the fundamental "order test" asks: "Does the order matter?" If yes, use permutations; if no, use combinations.

Answer: "Whether the order of the selected items matters." (option B).

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

A restaurant offers 6 types of sandwiches and 4 types of drinks. If a customer chooses one sandwich AND one drink, how many different meal combinations are possible?

A.
10
B.
24
C.
20
D.
15

Correct Answer:

Step-by-Step Solution

Key idea: When a task has independent stages connected by AND, use the multiplication principle. Step 1: Identify the stages: Stage 1 = choose a sandwich, Stage 2 = choose a drink. Step 2: Check independence: The choice of sandwich doesn't affect the choice of drink (and vice versa). Step 3: Count ways for each stage: - Sandwiches: 6 choices - Drinks: 4 choices Step 4: Apply multiplication principle: Total ways =

6 \times 4 = 24

. Answer: 24 (option B).

More notes in this unit

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Counting, Selections and Arrangements Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Counting, Selections and Arrangements notes for XAT: 14 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

What is the fundamental test to decide whether a counting problem requires permutations or combinations ?

Question 2

A restaurant offers 6 types of sandwiches and 4 types of drinks. If a customer chooses one sandwich AND one drink, how many different meal combinations are possible?

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