Problem 24 - Algebra for JEE ADVANCED

Question

The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is a. 27378 b. 27405 c. 27399 d. none of these

Step-by-Step Solution

Answer

The answer is 27,378.

Step 1: Understand the total selections

First, we calculate the total number of ways to select any four numbers from the list of numbers 1 to 30. This is a combinatorial problem where we can use the combination formula, often denoted as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 30 \) and \( r = 4 \), so the total number of ways is given by \( \binom{30}{4} \).

Step 2: Calculate total combinations

Calculate the total number of combinations using the formula:\[\binom{30}{4} = \frac{30 \times 29 \times 28 \times 27}{4 \times 3 \times 2 \times 1} = 27,405.\]This represents the total selections of four numbers from 1 to 30 without any restrictions.

Step 3: Understand exclusion of consecutive numbers

Next, we need to exclude selections that contain four consecutive numbers, like {1, 2, 3, 4} or {2, 3, 4, 5}. Recognize that any set of four consecutive numbers can be expressed as \( \{x, x+1, x+2, x+3\} \), where \( x \) is the starting number and must satisfy \( x \leq 27 \).

Step 4: Calculate consecutive selections to exclude

Count the number of ways to select four consecutive numbers. \( x \) ranges from 1 to 27, thus there are 27 such sets: \( \{1,2,3,4\} \), \( \{2,3,4,5\} \), ..., \( \{27,28,29,30\} \). Hence, 27 selections must be excluded from our total.

Step 5: Calculate valid selections

Subtract the excluded consecutive sets from the total number of combinations:\[27,405 - 27 = 27,378.\]These are the total number of ways to select four numbers from 1 to 30 that do not include any set of four consecutive numbers.

Key Concepts

Combination Formula

When approached with a problem of selecting a specific number of items from a larger pool, the combination formula becomes extremely handy. It is symbolically represented as \( \binom{n}{r} \), where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items you want to choose.

The combination formula helps you figure out how many different ways you can make these selections, while order is not important. It is calculated using the formula:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \( ! \) denotes factorial, which means the product of an integer and all the integers below it. In our problem, we wanted to select 4 numbers out of 30, which is calculated as \( \binom{30}{4} \). This means:

• We multiply the top four consecutive numbers starting from 30: 30, 29, 28, and 27.
• Then, divide by the product of 4 down to 1: 4, 3, 2, 1.

This calculation gives us the total number of ways to make an unrestricted selection.

Exclusion Principle

After calculating the unrestricted selections, we apply the exclusion principle to filter out unwanted sets. In our case, it is necessary to remove any selections that include four consecutive numbers. Selections such as \( \{1, 2, 3, 4\} \) or \( \{27, 28, 29, 30\} \) need to be excluded. To apply the exclusion principle:

• Identify patterns or criteria that selections must not satisfy (here, selecting four consecutive numbers).
• Count the number of such sets to exclude.

For four consecutive numbers, we can express them as \( \{x, x+1, x+2, x+3\} \), where \( x \) varies from 1 to 27. This means there are 27 sets of four consecutive numbers that need exclusion. Thus, we subtract these from the total combinations.

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order without any gaps. In a series of numbers, say from 1 to 30, a set of consecutive numbers can be \( \{5, 6, 7, 8\} \) or \( \{15, 16, 17, 18\} \). For this particular problem, identifying and excluding such sets is essential because they do not qualify as valid selections under the given constraints. Automated exclusion can occur by running through a sequence from the smallest possible starting number of 1 up to the highest that allows a full set—27 in this problem (since \( x+3 \leq 30 \)).To ensure all consecutive number sets are caught, we:

• Establish the smallest number, allowing a full set (1 for the starting number 1).
• End where the last consecutive set stops (27 for set starting at 27).
• Confirm there are exactly 27 such sets identified, aligning with our exclusion methodology.

Permutations and Combinations

Both permutations and combinations are key concepts in combinatorics, but they stand for different ideas:

• Permutations: It's about the arrangement where order does matter. For instance, arranging numbers 2, 3, and 1 can result in different orders like 231 or 312, each considered unique.
• Combinations: This involves selection where order does not matter. Combinations focus only on selecting items regardless of their sequence.

In the problem we're addressing, our focus is on combinations, as the order of selection does not change the outcome. Calculating through combinations gives the total possibilities, and further applying exclusions refines the result to meet the exercise requirements. Understanding the difference between these constructs is crucial. It ensures the correct method is employed in obtaining the solution. For exercises like ours, combinations offer the desired solutions by emphasizing selection over arrangement.