Question

How many different words can be made using the letters of the word 'HALLUCINATION' if all consonants are together? (a) 129780 (b) 1587600 (c) 35600 (d) none of these

Short answer

a) 50400 b) 100800 c) 126000 d) None of these Answer: d) None of these Solution: The total number of different arrangements of the word 'HALLUCINATION' with all consonants together is 151200, which isn't one of the given options.

Step-by-step solution

Identify vowels and consonants

First, we will identify the total number of consonants and vowels in the word "HALLUCINATION". The consonants are 'H', 'L' (which occurs twice), 'C', and 'N' (which occurs twice). The vowels are 'A' (which occurs thrice), 'U' (which occurs twice), and 'I' (which occurs twice).

Count distinct consonant and vowel groups

Since the consonants must be together, we can consider the consonant group 'HLCN' as a single unit. Now we have 3 vowel groups: one 'A' group (which occurs thrice), one 'U' group (which occurs twice), and one 'I' group (which occurs twice). So, we have a total of 3+1 = 4 groups.

Calculate arrangements within consonant and vowel groups

Now, let's calculate the arrangements within the consonant group and the vowel groups: 1. Consonant group: There are 5 consonants with 2 repetitions of 'L' and 2 repetitions of 'N', so the total arrangements within the consonant group are \frac{5!}{2! \cdot 2!} = 30. 2. Vowel groups: There are 7 vowels with 3 repetitions of 'A', 2 repetitions of 'U', and 2 repetitions of 'I', so the total arrangements within the vowel groups are \frac{7!}{3! \cdot 2! \cdot 2!} = 210.

Calculate the arrangements of groups

There are 4 groups in total (3 vowel groups + 1 consonant group). The total arrangements for these groups are 4! = 24.

Calculate the final result

Finally, we will find the total number of different arrangements of the word 'HALLUCINATION', with all consonants together by multiplying the arrangements of consonant group, vowel groups, and total groups: 30 × 210 × 24 = 151200. Since this isn't one of the given options, the correct answer is (d) none of these.

Key concepts

Arrangement of Letters

Understanding the arrangement of letters in permutations involves recognizing how to order a set of items uniquely. When we consider the word 'HALLUCINATION', it's like having a puzzle with each letter representing a different piece. To solve this puzzle, we need to determine how many unique pictures (words in this case) we can create.

The fundamental principle is straightforward: for every spot in the word, we have a choice of letters that could go there. We start by filling the first position, then the second, and so forth. However, when certain letters repeat, we must adjust our calculation. If 'A' appears three times, for example, arranging three 'A's doesn't create a new word—hence, we divide by the factorial of the number of repetitions to correct our count.

Using permutations allows us to consider all the possible ways we can arrange these letters, recognizing the distinction between vowels and consonants, especially when coupled with specific conditions such as grouping all consonants together as in our example.

Consonant Grouping

Grouping certain elements in permutation problems often changes the way we approach the solution. For consonant grouping, we treat all consonants as if they were a single 'super-letter' or unit. This simplifies our calculation since we now need to arrange fewer groups rather than each individual consonant.

In our example with 'HALLUCINATION', once we form the single unit with the consonants 'HLCN', it becomes one item in our list to permute, along with the distinct vowel groups. This effectively reduces the task of arranging many individual letters into a more manageable task of arranging fewer groups. However, within the consonant group, each letter's distinct arrangement still needs to be considered, which involves further permutations where repetitions are also accounted for.

Factorial Calculation

A factorial, denoted by an exclamation point (!), is a function that multiplies a series of descending natural numbers. For example, the factorial of 5 (written as 5!) is calculated as 5 x 4 x 3 x 2 x 1, which equals 120. Factorials are immensely useful in permutation and combination problems because they represent the total number of ways a set number of items can be arranged.

In the realm of permutations, we use factorials to find out how many different ways we can organize a group. When a problem mentions that there are repetitions, as with the letters in 'HALLUCINATION', we divide the factorial by the factorials of the number of repeats to ensure we aren't overcounting arrangements that are essentially identical. This forms the basis for calculating arrangements accurately, regardless of complexity.

Repetition of Letters

The repetition of letters within a word adds a layer of complexity to permutation problems. When letters repeat, like 'L' and 'N' in 'HALLUCINATION', simply calculating the total number of arrangements without adjustment would inflate our count, as many arrangements would look identical.

To address this, we divide by the factorial of the number of times each letter repeats. This process, known as adjusting for indistinguishable permutations, ensures that we're counting only the unique arrangements. It is as if we're saying, 'While there are several 'L's, swapping them doesn't create a new order; they are indistinguishable.' By incorporating the division of factorials for repeats, we maintain the integrity of our arrangement count, reflecting only the genuinely distinct possibilities.