We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can

NCERT Exemplar · Q5

6

Punjab PsebShort· 3mImportance★★★★★est

55% · 71/130 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Permutations Without Repetition

Permutations Without Repetition – The Idea of Arranging Things

Imagine you have three different books on a shelf: a Physics book, a Chemistry book, and a Maths book. How many different ways can you arrange them in a row?

You could try listing them out:

Physics, Chemistry, Maths
Physics, Maths, Chemistry
Chemistry, Physics, Maths
Chemistry, Maths, Physics
Maths, Physics, Chemistry
Maths, Chemistry, Physics

That's 6 arrangements. Notice that each arrangement uses all three books exactly once — no book is repeated, and no book is left out. This is the core idea: permutations without repetition count the number of ways to arrange a set of distinct objects in order, using each object exactly once.

Why "Without Repetition"?

The phrase "without repetition" means that once you place an object in a position, you cannot use it again. In our book example, once you put the Physics book in the first slot, you cannot put it in the second or third slot. Each object appears exactly once in the arrangement.

This is different from "permutations with repetition" (like creating 3-letter codes from the letters A, B, C where you can reuse letters — e.g., AAA, AAB, etc.). Here, no repeats allowed.

The Counting Logic – Why Multiply?

Let's build the arrangement step by step for 3 distinct books:

First position: You have 3 choices (any of the 3 books).
Second position: After placing the first book, only 2 books remain — so 2 choices.
Third position: Only 1 book is left — so 1 choice.

Total arrangements = $3 \times 2 \times 1 = 6$ .

This product $3 \times 2 \times 1$ is called 3 factorial, written as $3!$ .

P(n) = n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1

For $n$ distinct objects, the number of permutations (arrangements in order) is $n!$ .

What If You Only Arrange Some of Them?

Suppose you have 5 different books, but you only want to arrange 3 of them on a shelf. How many ways?

First position: 5 choices
Second position: 4 choices
Third position: 3 choices

Total = $5 \times 4 \times 3 = 60$ .

This is a permutation of 5 objects taken 3 at a time, written as $P(5,3)$ or ${}^5P_3$ .

P(n, r) = \frac{n!}{(n-r)!} = n \times (n-1) \times \cdots \times (n-r+1)

Here $n$ is the total number of distinct objects, and $r$ is how many you are arranging. The formula works because:

Numerator $n!$ counts all arrangements of all $n$ objects.
Denominator $(n-r)!$ removes the arrangements of the $n-r$ objects you are not using.

Key Points to Remember

Important

Order matters — swapping two objects gives a different permutation.
No repetition — each object is used at most once.
For arranging all $n$ objects: $n!$
For arranging r out of $n$ objects: $\frac{n!}{(n-r)!}$

Common Mistake to Avoid …

Start your 14-day free trial to unlock the full solution →

The problem requires selecting 6 persons from 8, with the condition that if person A is chosen, person B must also be chosen. This is solved by considering two mutually exclusive cases: either A is chosen (which forces B to be chosen), or A is not chosen. The total number of ways is $\boxed{22}$ .

This problem asks us to find the number of ways to select a group of people, which means the order of selection does not matter. This is a classic combination problem. The twist is the conditional constraint: "if person A is chosen, then B must be chosen." This condition dictates how we approach the selection process.

The core idea is to break down the problem into distinct, non-overlapping scenarios (cases) that cover all possibilities while respecting the given condition. The condition "if A is chosen, then B must be chosen" can be rephrased as: we cannot have A in our group without B also being in our group.

Let's consider the implications of this condition by focusing on person A:

Case 1: Person A is chosen.

If A is chosen, the condition states that B must also be chosen. This means A and B are both definitely part of our selected group of 6 persons.
Case 2: Person A is NOT chosen.

If A is not chosen, the premise of the condition ("A is chosen") is false. In logic, if the premise of an "if-then" statement is false, the entire statement is considered true (vacuously true). This means if A is not chosen, there is no restriction on B; B can either be chosen or not chosen. We simply proceed with selecting 6 people from the remaining pool, excluding A.

These two cases are mutually exclusive (A cannot be both chosen and not chosen simultaneously) and exhaustive (they cover all possibilities for A). Therefore, we can find the number of ways for each case and add them up.

We are selecting $6$ persons from a total of $8$ .

Calculate the number of ways if A is chosen.
- If A is chosen, then B must also be chosen.
- This means we have already selected 2 persons (A and B) for our group of 6.
- We need to select $6 - 2 = 4$ more persons.
- The original pool of 8 persons now has A and B removed (since they are already selected). So, the remaining pool from which we can choose is $8 - 2 = 6$ persons.
- The number of ways to choose these 4 persons from the remaining 6 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ . …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ4 — Out of 18 points in a plane, no three are in the same line except five…→Next questionQ6 — How many committee of five persons with a chairperson can be selected …

Q.We wish to select 666 persons from 888, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?

🎓Unlock everything free for 14 days

Q.We wish to select $6$ persons from $8$ , but if the person A is chosen, then B must be chosen. In how many ways can selections be made?

Unlock everything free for 14 days