Question

If all vowels occupy odd places, how many words can be formed from the letters of the word HALLUCINATION? (a) 129650 (b) 1587600 (c) 78500 (d) none of these

Short answer

Answer: (d) none of these (The total number of ways is 7,257,600)

Step-by-step solution

Determine the number of odd and even places

The given word "HALLUCINATION" has a total of 13 letters. We have 1, 3, 5, 7, 9, 11, and 13 as odd places, and 2, 4, 6, 8, 10, and 12 as even places. In total, there are 7 odd places and 6 even places.

Count vowels and consonants

Now we need to count the number of vowels and consonants in the given word. The word HALLUCINATION has the following vowels and consonants: Vowels: A (2), I (2), O (1), U (1) - Total: 6 vowels Consonants: H (1), L (2), N (1), T (1) - Total: 5 consonants

Arrange vowels in odd places

We have 6 vowels and 7 odd places. We need to arrange 6 vowels in 7 odd places. The number of possibilities for this arrangement can be calculated using permutations as: P(7,6) = \frac{7!}{(7-6)!} = 7 * 6 * 5 * 4 * 3 * 2

Arrange consonants in even places

We have 5 consonants and 6 even places. We need to arrange 5 consonants in 6 even places. The number of possibilities for this arrangement can be calculated using permutations as: P(6,5) = \frac{6!}{(6-5)!} = 6 * 5 * 4 * 3 * 2

Calculate total word formations

Now to find the total number of word formations, we need to multiply the number of possibilities of vowels with the number of possibilities of consonants: Total word formations = P(7,6) * P(6,5) = (7 * 6 * 5 * 4 * 3 * 2) * (6 * 5 * 4 * 3 * 2) = 10,080 * 720 = 7,257,600 Since the result of 7,257,600 does not match any of the given options, the answer is: (d) none of these

Key concepts

Vowels and Consonants

In the English language, letters are categorized into vowels and consonants. Vowels include the letters A, E, I, O, and U. Consonants consist of the remaining letters of the alphabet, except for Y, which sometimes acts as both.
In the context of word formation problems, it is crucial to distinguish between vowels and consonants to apply specific rules governing their arrangement. For instance, vowels may be required to occupy specific positions, such as odd-numbered places in a sequence, which can significantly influence the total number of permutations possible.
In the word "HALLUCINATION," we identify six vowels: two A's, two I's, one O, and one U. The remaining five letters—H, L, L, N, and T—are consonants. Separating vowels and consonants helps simplify complex word arrangements. Moreover, it facilitates the application of permutation formulas, essential in solving numerical problems related to these categories.

Word Formation

Word formation involves arranging a set of letters to form words, observing specific conditions or constraints.
This concept is critical in both linguistic studies and quantitative problems. In exercises such as the one involving the word "HALLUCINATION," letters must be arranged in compliance with specific rules, such as placing vowels in odd-numbered positions.
To calculate word formations, permutations are used. These mathematical concepts help determine the number of ways to arrange items where order is important. For example, in arranging vowels in seven available odd positions, the permutation formula, denoted as \( P(n, r) = \frac{n!}{(n-r)!} \), is applied, where \( n \) is the total positions and \( r \) is the number of vowels. Thus, understanding word formation requires grasping how to apply these principles to different constraints and scenarios effectively.

Quantitative Aptitude

Quantitative aptitude is a branch of mathematics focused on solving numerical problems using logical reasoning and mathematical concepts. It is a critical component in exams and assessments to evaluate problem-solving and analytical skills.
In the context of permutations and word formations, quantitative aptitude involves calculations that test an individual's ability to discern patterns and solve complex arrangements under given conditions. This may include determining the possible arrangements of consonants and vowels in a sequence when certain constraints are applied, such as vowels occupying only odd positions in a word configuration.
Exercises like arranging letters based on specific rules utilize core mathematical concepts, such as factorials and permutations, which are pivotal to quantitive reasoning. By mastering these calculations, students can successfully navigate many quantitative aptitude problems, often seen in competitive exams or job assessments.