Problem 64 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of permutations of letters \(a, b, c, d, e, f, g\) so that neither the pattern beg nor cad appears is (A) \(\frac{7 !}{3 ! 3 !}\) (B) \(\frac{7 !}{2 ! 3 ! 3 !}\) (C) 4806 (D) None of these

Step-by-Step Solution

Answer

The answer is 3636, which matches option (D) None of these.

Step 1: Understanding the Problem

We are asked to find the number of permutations of the 7 letters where neither of the patterns 'beg' nor 'cad' appears.

Step 2: Calculating Total Permutations

First, calculate the total number of permutations of the 7 letters. This can be found using the formula for permutations, which is the factorial of the number of items: \[ 7! = 5040 \]

Step 3: Counting Permutations for Each Pattern

Now, consider the letters 'beg' together as a single block. This means we have 5 blocks: 'beg', 'a', 'c', 'd', and 'f.' The number of ways to arrange these blocks is:\[ 5! = 120 \]Within the 'beg' block, the letters 'b', 'e', 'g' can be permuted as:\[ 3! = 6 \]Thus, the permutations containing 'beg' are:\[ 5! \times 3! = 720 \]

Step 4: Counting Permutations for the Other Pattern

Similarly, treat 'cad' as a block, giving us 5 blocks: 'cad', 'b', 'e', 'f', and 'g.' The number of arrangements of these blocks:\[ 5! = 120 \]The internal arrangement of 'cad' is:\[ 3! = 6 \]The permutations containing 'cad' are:\[ 5! \times 3! = 720 \]

Step 5: Subtracting Overcounts

Subtraction of the double-counting is necessary for permutations that contain both 'beg' and 'cad'. Treat both 'beg' and 'cad' as two blocks along with 'f'. Thus, we have:\[ 3! = 6 \]Additionally, within each block, permutations of 'beg' and 'cad':\[ 3! \times 3! = 36 \]So, permutations within 'beg' or 'cad' is:\[ 3! \times 3! \times 1! = 36 \]

Step 6: Applying the Inclusion-Exclusion Principle

Subtract permutations with 'beg' and 'cad' using the inclusion-exclusion principle:\[ \text{Permutations without 'beg' or 'cad'} = 7! - (\text{Permutations with 'beg'} + \text{Permutations with 'cad'} - \text{Overcounts of both 'beg' and 'cad'}) \]\[ = 5040 - (720 + 720 - 36) \]\[ = 5040 - 1404 \]\[ = 3636 \]

Step 7: Final Verification and Calculation

The calculations match option (D) None of these, as the correct number of permutations where neither 'beg' nor 'cad' occurs is 3636.

Key Concepts

Factorials: Understanding Their Role in Permutations

Factorials are essential in the realm of permutations. They help us quickly calculate the number of ways to arrange a set of items where order matters. The factorial function, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 7! \) (read as "seven factorial") equals \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).

• Factorials grow very rapidly with increasing \( n \). This makes them especially useful in calculating large permutation possibilities effectively.
• In the exercise problem, \( 7! \) represents the total number of permutations of seven different letters. Thus, it quantifies all possible arrangements of the letters a through g.

Whenever dealing with permutations, always start by calculating the factorial of the total number of elements to lay the foundation for more intricate calculations later on.

Inclusion-Exclusion Principle: Avoiding Double-Counting

The inclusion-exclusion principle is a clever method used in combinatorics to count the elements in the union of multiple sets without overcounting. When two or more sets can overlap (or partially contain common elements), it ensures accuracy in counting by subtracting the overcounted parts.

• In this exercise, we want to avoid the arrangements containing 'beg' or 'cad' as patterns in the permutations of letters.
• The principle helps remove misconceptions by correcting for overlaps when adding permutations containing 'beg' and 'cad'. We first add up each distinct permutation count and then subtract the overlapping permutation count.

Applying this principle reduces errors, whether we’re dealing with physical arrangements or abstract elements like these patterns, ensuring accurate permutations counts.

Combinatorics: Solving Permutation Challenges

Combinatorics is a branch of mathematics focused on counting, arrangements, and combinations. It provides us with the rules and formulas to solve complex arrangement problems such as the permutations seen in our exercise.

• Understanding combinatorics allows us to methodically tackle challenges by breaking down problems into smaller, more manageable parts.
• Our exercise required identifying the patterns 'beg' and 'cad' and treating them as single units to simplify the permutation possibilities.
• By applying tools like factorials and the inclusion-exclusion principle, combinatorics helps derive the correct number of permutations systematically.

With the right techniques and a clear understanding of concepts like permutation and combination, combinatorics offers powerful solutions to otherwise complicated counting problems.