Problem 66 - Algebra for JEE ADVANCED

Question

The number of different ways in which five 'alike dashes' and eight 'alike dots' can be arranged using only seven of these "dashes' and 'dots' is a. 350 b. 120 c. 1287 d. none of these

Step-by-Step Solution

Answer

1716 arrangements.

Step 1: Understand the Problem

We need to find the number of different ways to arrange a total of 7 items, choosing from 5 alike "dashes" and 8 alike "dots".

Step 2: Determine Total Arrangements

To find the number of ways to select a total of 7 items (dashes + dots), we consider that we are making combinations of 7 selections from a total of 13 available selections (5 dashes + 8 dots). Hence, we must use combinations formula.

Step 3: Use Combinations Formula

The formula to find combinations is given by \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of selections to make. Therefore, \[ \binom{13}{7} = \frac{13!}{7!(13-7)!} = \frac{13!}{7!6!} \].

Step 4: Calculate Factorials and Solve

Calculate the factorial values: \( 13! = 6227020800 \), \( 7! = 5040 \), and \( 6! = 720 \). Plug these into the combinations formula: \[ \binom{13}{7} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 1716 \].

Step 5: Consider Both Dashes and Dots

Given the problem constraints, for any specific number of "dashes", we can find the number of ways to choose where these dashes appear among the 7 items. Sobtract from total arrangements for dashes or dots exceeding total selections.

Step 6: Final Answer

Based on the calculated combinations, the total number of ways to arrange seven of these items abiding by the constraints is \( 1716 \).

Key Concepts

Combinations Formula

When faced with a problem of selecting a certain number of items from a larger group, we use the concept of combinations. This involves selecting items where the order does not matter. The combinations formula is written as \( \binom{n}{r} \), indicating the number of ways to choose \( r \) items from a total of \( n \) items.

For our problem, we want to arrange 7 items consisting of 'dashes' and 'dots' out of a total of 13. The formula used is:

• \( \binom{13}{7} = \frac{13!}{7!(13-7)!} \)

This formula helps in calculating the possible combinations of choosing 7 items from 13.

The use of this formula is crucial in problems involving combinations, where a specific arrangement or ordering of items is not relevant.

Factorial Calculations

Factorials play a key role when dealing with permutations and combinations.

A factorial, represented by an exclamation mark \(!\), refers to the product of an integer and all the integers below it. For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)

In our original exercise, we computed \( 13! \), \( 7! \), and \( 6! \) as part of applying the combinations formula. Factoring these into \( \binom{13}{7} \) gives us huge numbers, but when divided by each other, they simplify to manageable proportions.

A basic understanding of factorial calculations helps you tackle larger problems by simplifying calculations in both combinations and permutations.

Arrangement Problems

Arrangement problems often involve deciding how to organize a set of items according to specific rules or constraints. In combinatorics, these arrangements can be divided into problems where either the order matters (permutations) or it does not (combinations).

In the exercise example, the focus was on combinations, suggesting that the order does not matter. We were asked to determine the number of ways to arrange 7 items from a set group of indistinguishable 'dashes' and 'dots'. The uniqueness, or lack thereof, of these items affects the calculation.

This type of arrangement problem is extremely common in real-world scenarios, such as selecting a committee from a large group of people or deciding on a subset of features for a product. Understanding different types of arrangement problems and applying appropriate formulas, like combinations, is essential for accurate outcomes.