Problem 38 - Algebra for JEE ADVANCED

Question

The number of ways in which ten candidates \(A_{1}, A_{2}, \ldots, A_{10}\) can be ranked such that \(A_{1}\) is always above \(A_{10}\) is a. \(5 !\) b. \(2(5 !)\) c. \(10 !\) d. \(\frac{1}{2}(10 !)\)

Step-by-Step Solution

Answer

The number of ways is \(\frac{1}{2}(10!)\), or 181440.

Step 1: Understanding the Total Arrangements

To solve this problem, first understand the total number of ways to arrange 10 candidates in any order. The number of permutations of 10 items is given by the factorial of 10, which is \[ 10! \] Calculating this gives:\[10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880\]

Step 2: Identify the Condition

The problem requires candidate \(A_1\) to always be ranked higher than candidate \(A_{10}\). This splits each of the total arrangements into two possible groups: one where \(A_1\) is above \(A_{10}\) and one where \(A_{10}\) is above \(A_1\).

Step 3: Divide by Possible Arrangements

Since exactly half of any arbitrary arrangement of \(A_1\) and \(A_{10}\) will have \(A_1\) above \(A_{10}\) (and the other half will have \(A_{10}\) above \(A_1\)), the number of arrangements where \(A_1\) is ranked above \(A_{10}\) can be found by dividing the total arrangements by 2:\[\text{Number of desired rankings} = \frac{362880}{2} = 181440\]

Step 4: Confirm the Answer

The solution matches exactly with the choice where the total arrangements are divided into two equal parts, as calculated:\[ \frac{1}{2}(10!) = 181440 \]Thus, the correct answer is option \(d\).

Key Concepts

Factorial

Factorial is a fundamental concept in mathematics, especially when dealing with permutations and arrangements. It's represented by an exclamation mark after a number, like this: 10! (pronounced "ten factorial"). The factorial of a number is the product of all positive integers up to that number. For example: - The factorial of 3 (3!) is 3 × 2 × 1 = 6. - The factorial of 5 (5!) is 5 × 4 × 3 × 2 × 1 = 120. In problems involving rankings or arrangements of elements, factorials help calculate the total number of possible sequences. This is because each additional element multiplies the potential arrangements by that many more options. The larger the number, the more exponentially the arrangements grow, which is why factorials are great for capturing all possibilities in permutations.

Arrangements

Arrangements refer to the different ways you can order or position a set of items. When we talk about arranging, we think about how we can line these items up in a sequence or order. Consider 10 candidates you need to arrange. The total number of ways you can arrange these 10 individuals is a permutation, calculated by the factorial of 10, or 10!. This means there are 3,628,800 possible sequences for arranging these 10 candidates. However, if additional conditions are applied to these arrangements (like specific individuals being above or below others), it splits the total arrangements into separate categories. For instance, having one candidate always ranked above another modifies the arrangement count because we must exclude contrary arrangements.

Ranking

Ranking involves assigning a position or order to items based on specific criteria. In our problem, we rank 10 candidates, ensuring one specific arrangement condition: candidate A1 must be ranked higher than A10. This constraint leaves us with two ranking scenarios for any sequence:

• A1 is ranked above A10.
• A10 is ranked above A1.

Each of these scenarios includes half of the total possible arrangements. Why? Because in any arrangement of A1 and A10, you will have a 50% chance of one being above the other. By appreciating this division, we understand why the total number of applicable rankings is half of the complete arrangements, giving us the desired permutation count.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects belonging to a certain set, often under certain constraints or conditions. It plays a crucial role in problems related to permutations, as it helps in understanding how items can be sorted, selected, or arranged. In our exercise, combinatorics comes into play when identifying both permutations and conditions placed on these permutations. By knowing that each item can potentially occupy every position, yet accounting for certain conditions like "A1 must be above A10," combinatorics allows us to calculate how combinations are affected by such rules. Using combinatorics, we understood why the total permutations are reduced by half to satisfy the given constraint, efficiently solving permutation problems with added conditions.