Counting, Selections and Arrangements Practice Questions for XAT: 69+ Solved Questions with Step-by-Step Solutions

Solve 69+ Counting, Selections and Arrangements practice questions for XAT with answers and detailed solutions. Free sample questions below.

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.

Counting, Selections and Arrangements: Solved Questions with Step-by-Step Explanations (5 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).

Question 3 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

In how many ways can you select 2 students from a group of 5 students to form a study pair? (The order of selection does not matter.)

A.
10
B.
20
C.
5
D.
25

Correct Answer:

Step-by-Step Solution

Key idea: When order does not matter and we're selecting

r

items from

n

items, use the combination formula. Step 1: Identify the parameters: - Total students (

n

) = 5 - Students to select (

r

) = 2 - Order doesn't matter → combination Step 2: Apply the combination formula:

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

Step 3: Substitute values:

^{5} C_{2} = (2 5) = \frac{5 !}{2 ! ( 5 - 2 )!} = \frac{5 !}{2 ! \cdot 3 !}

Step 4: Calculate:

\frac{5 \times 4 \times 3 !}{2 \times 1 \times 3 !} = \frac{5 \times 4}{2} = \frac{20}{2} = 10

Answer: 10 (option A).

Question 4 · Quantitative Aptitude and Data Interpretation (QA & DI) MSQ

From a box containing 5 distinct red balls and 4 distinct blue balls, 4 balls are to be selected such that there are at least 2 red balls. Which of the following expressions correctly represent the number of ways to make this selection?

A.
$(2 5) \times (2 7)$
B.
$(2 5) (2 4) + (3 5) (1 4) + (4 5)$
C.
$(4 9) - (4 4) - (1 5) (3 4)$
D.
$(2 5) (2 4) + (3 5) (1 4) + (4 5) (0 4)$

Correct Answer:

["B","C","D"]

Step-by-Step Solution

Key idea: This is an "at least" selection problem. The key discrimination is recognising the classic overcounting trap in the direct method and verifying the correct casework and complement expressions.

Step 1: Evaluate Option A: $(2 5) \times (2 7)$ .

This method chooses 2 red balls first, then 2 from the remaining 7. This overcounts cases where 3 or 4 red balls are selected, because the same final set of balls can be formed by choosing different "initial" 2 red balls. This is incorrect.

Step 2: Evaluate Option B: $(2 5) (2 4) + (3 5) (1 4) + (4 5)$ .

This is the correct casework method. It sums the mutually exclusive cases: exactly 2 red (and 2 blue), exactly 3 red (and 1 blue), and exactly 4 red (and 0 blue). This is correct.

Step 3: Evaluate Option C: $(4 9) - (4 4) - (1 5) (3 4)$ .

This is the correct complement method. Total ways $(4 9)$ minus the invalid cases: 0 red (4 blue) and 1 red (3 blue). This is correct.

Step 4: Evaluate Option D: $(2 5) (2 4) + (3 5) (1 4) + (4 5) (0 4)$ .

This is identical to Option B, just explicitly writing $(0 4)$ for the last term. This is correct.

Answer: B, C, D

Question 5 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

In a survey of 50 people, 25 like product A, 22 like product B, and 20 like product C. It is known that 8 people like both A and B, 7 like both B and C, and 6 like both A and C. If $x$ people like all three products, what is the maximum possible value of $x$ ?

A.
2
B.
4
C.
6
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is an overlapping categories (Venn diagram) problem requiring the inclusion-exclusion principle and inequality bounds to find the maximum intersection.

Step 1: Let $N$ be the total number of people (50). Let $n (A \cup B \cup C)$ be the number of people who like at least one product, and $N o n e$ be the number who like none.

Step 2: Apply the inclusion-exclusion formula:

$n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (A \cap C) + n (A \cap B \cap C)$

Step 3: Substitute the given values:

$n (A \cup B \cup C) = 25 + 22 + 20 - 8 - 7 - 6 + x = 46 + x$ .

Step 4: We know that $n (A \cup B \cup C) + N o n e = 50$ .

Therefore, $(46 + x) + N o n e = 50$ , which simplifies to $x + N o n e = 4$ .

Step 5: Since the number of people who like none ( $N o n e$ ) cannot be negative, $N o n e \geq 0$ .

To maximize $x$ , we must minimize $N o n e$ . The minimum value for $N o n e$ is 0.

Step 6: If $N o n e = 0$ , then $x = 4$ . We must verify that no individual region in the Venn diagram becomes negative for $x = 4$ .

Only A = $25 - 8 - 6 + 4 = 15 \geq 0$ .

Only B = $22 - 8 - 7 + 4 = 11 \geq 0$ .

Only C = $20 - 7 - 6 + 4 = 11 \geq 0$ .

All regions are non-negative, so $x = 4$ is valid.

Answer: 4

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?

Question 3

In how many ways can you select 2 students from a group of 5 students to form a study pair? (The order of selection does not matter.)

Question 4

From a box containing 5 distinct red balls and 4 distinct blue balls, 4 balls are to be selected such that there are at least 2 red balls. Which of the following expressions correctly represent the number of ways to make this selection?

Question 5

In a survey of 50 people, 25 like product A, 22 like product B, and 20 like product C. It is known that 8 people like both A and B, 7 like both B and C, and 6 like both A and C. If x people like all three products, what is the maximum possible value of x ?

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Counting, Selections and Arrangements Practice Questions for XAT: 69+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Counting, Selections and Arrangements

Chapter Roadmap

1. Fundamental Counting Principle

2. Permutations: Art of Arrangement

3. Combinations: Art of Selection

4. Grouping and Distribution

5. Advanced Constraints

The Core Intuition: And vs Or

The Core Intuition: AND vs OR

Multiplication (AND)

Addition (OR)

Counting, Selections and Arrangements: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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