Problem 42 - Algebra for JEE ADVANCED

Question

Among 10 persons, \(A, B, C\) are to speak at a function. The number of ways in which it can be done if \(A\) wants to speak before \(B\) and \(B\) wants to speak before \(C\) is a. \(10 ! / 24\) b. \(9 ! / 6\) c. \(101 / 6\) d. none of these

Step-by-Step Solution

Answer

Correct answer is not present in the given options.

Step 1: Understand the Problem

We need to find the number of ways in which 3 specific people among 10 will speak, following a particular order: A before B, and B before C.

Step 2: Recognize Order Constraints

Since A must speak before B, and B must speak before C, the order of these 3 particular speakers is fixed. Thus, there is exactly 1 specific order among any grouping of these 3 people.

Step 3: Calculate Total Arrangements

Calculate the total number of ways to arrange 10 people in a sequence, which is given by the factorial: \(10!\).

Step 4: Factor in Order Constraint Among A, B, C

Given A must be before B, and B before C, consider that amongst any arrangements of people, there are 6 possible sequences for A, B, and C (as any of A, B, C can interchange places). This includes sequences like ABC, ACB, BAC, etc. With only ABC desirable, only 1/6th of all permutations satisfy the condition.

Step 5: Final Calculation

Divide the total arrangements by 6 to account for the order constraint: \( \frac{10!}{6} \). On evaluating, find \(10! = 3628800\), and hence : \( \frac{3628800}{6} = 604800 \).

Step 6: Verify Answer Among Options

Since \(9! = 362880\), \(\frac{9!}{6} = 60480\), the answer is not matching any given choice directly but aligns with the logic of one-third as derived initially.

Key Concepts

Factorial Calculation

Factorials are a fundamental concept in permutations and combinations. They serve as a cornerstone when calculating the number of ways objects can be arranged in a sequence. A factorial, noted as \( n! \), represents the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In the given problem, we calculate the total number of ways to arrange 10 people using a factorial. Thus, \( 10! \) denotes all possible arrangements of all 10 speakers provided there were no restrictions. Plugging into the equation, \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \). This large number signifies just how many different sequences can occur when order is not restricted.

Order Constraints

Order constraints specify conditions about the sequence in which certain events must occur. In permutations, this often translates to a fixed positional arrangement within a larger context. For our problem, the constraint dictates that the individuals A, B, and C must speak in that set order: A first, followed by B, then C.

This introduces a condition which reduces the total possibilities. Initially, without considering an order, the group A, B, C could be arranged in several ways, such as ABC, ACB, BAC, etc., precisely 6 ways. When we impose the constraint, only the sequence ABC is acceptable.

• ABC: Desired arrangement
• ACB, BAC, BCA, CAB, CBA: Unwanted arrangements

Thus, among any arrangements that include A, B, and C, only one in six will satisfy the condition, reducing possibilities to a fraction \( \frac{1}{6} \). This constraint ensures only valid sequences are considered in the final permutation count.

Arrangements in Sequence

Once a sequence is established, calculating valid arrangements involves considering both unrestricted and restricted permutations. The goal is to determine how a subset of elements in a fixed order integrates within a larger set.

In our exercise, the number of valid ways to incorporate the fixed sequence of A, B, C among the 10 individuals is achieved through initial factorial calculation, then adjustment for the imposed sequence order. First, identify arrangements without restrictions: \( 10! \).

• Total unrestricted permutations: \( 10! \)
• Account for fixed sequence ABC: \( \frac{10!}{6} \)

This simplifies to:\[\frac{3,628,800}{6} = 604,800\]

Thus, there are 604,800 possible ways to arrange 10 people, adhering to the fixed speaking order. Understanding sequence arrangements aids in addressing problems where specific items or events must appear before others within a larger set. This adjustment ensures the solution aligns with the given constraints, accurately reflecting reality.