Problem 23 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If the letters of the word REGULATIONS be arranged at random. What is the chance that there will be exactly 4 letters between the \(R\) and the \(E ?\) (a) \(6 / 55\) (b) \(8 / 55\) (c) \(10 / 55\) (d) \(12 / 55\)

Step-by-Step Solution

Answer

The chance is \(\frac{6}{55}.\)

Step 1: Count total letters

The word REGULATIONS has 11 different letters. We need to find how many ways we can arrange these letters.

Step 2: Calculate total possible arrangements

The total number of ways to arrange 11 different letters is given by the factorial of 11. Therefore, total arrangements = 11! = 39,916,800.

Step 3: Fix positions for R and E

We want exactly 4 letters between the letters R and E. Consider R is placed in a position, say position 1, then E must be at position 6. The letters between them can have any 4 letters from the remaining 9 letters.

Step 4: Count total valid positions for R and E

The pair (R, E) or (E, R) can be positioned in the following slots: (1, 6), (2, 7), (3, 8), (4, 9), (5, 10), (6, 11). This provides 6 possible starting positions for R, with E placed 4 slots down from R.

Step 5: Arrange the other letters

For each valid pair of positions for R and E, the 4 letters in between them can be selected and arranged in 4! ways from the 9 available letters, and the remaining 5 letters can be arranged in 5! ways. Thus, arranging all letters is 6 * 4! * 5! = 17,280 ways.

Step 6: Calculate probability

To find the probability, divide the number of favorable outcomes by the total possible arrangements: \(\frac{17,280}{39,916,800} = \frac{6}{55}.\)

Key Concepts

Factorials

Factorials are a key mathematical concept used to determine the total number of ways to arrange a set of distinct items. It is denoted by an exclamation mark \(!\) following a number. For example, \(11!\) represents the factorial of 11, which is the product of all positive integers from 11 down to 1.
In the context of the problem, calculating the factorial of the number of letters in the word REGULATIONS helps us determine how many different ways these letters can be arranged. Specifically:

• \(11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800\)

This factorial calculation is fundamental as it provides a baseline for understanding all possible arrangements of the letters when attempting to find specific arrangements, such as those with certain letters between them.

Permutations

Permutations are arrangements of objects in which order matters. In the problem, we are interested in specific permutations where there are exactly 4 letters between the letters R and E.
Unlike combinations, permutations give value to the sequence in which the objects appear. In this case, the positions where R and E can be placed are specific slots that leave exactly 4 spaces between them. The goal is to identify these permutations among all possible arrangements:

• Choosing positions like \((1, 6)\) or \((2, 7)\)
• Recognizing that for each positional pair, the arrangement of letters between them creates distinct permutations.

This means that for each placement of R, there is a corresponding position for E where exactly 4 letters fit between them, making each unique sequence or permutation significant to the solution.

Arrangement of letters

The arrangement of letters is central to solving the problem. The sequence in which letters are placed determines whether there are exactly 4 letters between R and E.
To break it down:

• Identify potential slots for fixing R and E, ensuring a 4-letter separation (e.g., \((1, 6), (2, 7), ...\))
• Calculate the arrangements for the letters in between R and E, using \(4!\), since any 4 letters from the remaining 9 can fill these slots
• Arrange the rest of the letters using \(5!\)

Combining these elements provides a complete picture of how to manage letter arrangements to meet the given condition. Important to note is that each potential slot combination contributes uniquely to the total number of favorable arrangements.

Combinatorics

Combinatorics encompasses the tools and methods for calculating outcomes of arrangements and combinations, a field under which both factorials and permutations fall.
In this problem, combinatorics involves selecting, arranging, and calculating favorable outcomes. The steps include:

• Using factorial to determine all possible arrangements of the letters (\(11!\))
• Finding allowable combinations for positioning R and E followed by permutations of the interior letters \(4!\) and the remaining letters \(5!\)

Finally, combinatorial probability is used to determine the likelihood that R and E will have exactly 4 letters between them, dividing the favorable outcomes by the total arrangements. Here, we applied \(\frac{17,280}{39,916,800}\) which simplifies to the probability \(\frac{6}{55}\). This showcases the application of combinatorics to handle complex arrangements and probabilities efficiently.