Exploring Probabilities with the Word 'FRACTION'
This article delves into the fascinating world of arrangements and probabilities, focusing specifically on the word 'FRACTION'. We will explore the different ways to arrange the letters of this word, particularly with a focus on starting and ending with vowels. Understanding these concepts can provide a rich insight into the intricacies of permutations and combinations in probability theory.
Understanding the Word 'FRACTION'
The word 'FRACTION' is composed of 8 letters. Among these letters, 3 are vowels: A, I, and O. This set-up creates an interesting scenario for examining the probability of arranging the word in such a way that it starts and ends with a vowel.
Combinations of Vowels
Firstly, let us consider the methods for arranging the vowels within the word. The three vowels (A, I, and O) can be arranged in different ways. The number of ways to arrange 3 distinct items is given by the factorial of 3, which is 3! (3 factorial). Mathematically, this is expressed as:
3! 3 × 2 × 1 6
Hence, there are 6 different ways to arrange these 3 vowels.
Arranging the Consonants
Now, let's consider the six consonants in the word 'FRACTION', which are F, R, C, T, N, and T. These consonants can also be arranged in many different ways. The total number of permutations of these 6 letters, taking into account that the letter T appears twice, is given by:
6! / 2! 720 / 2 360
Therefore, the 6 consonants can be arranged in 360 different ways.
Total Arrangements with Vowels at the Ends
A crucial aspect of this problem is the requirement for the word to start and end with a vowel. We will consider the two situations: the first vowel can be any of the 3, and the last vowel can be any of the remaining 2 vowels after the first one. The total number of arrangements that fit this criterion can be calculated as follows:
6 vowels × 360 consonant arrangements 2,160
Thus, there are 2,160 distinct ways to arrange the letters of 'FRACTION' such that the arrangement starts and ends with a vowel.
Probability Calculation
With the total number of possible arrangements being 4,320 (as shown in the initial computation), the probability of arranging 'FRACTION' in such a way that it starts and ends with a vowel can be calculated as:
(2,160 / 4,320) 1/2
Therefore, the probability that the word 'FRACTION' starts and ends with a vowel is 1/2 or 50%.
Conclusion
In summary, understanding the probability that the word 'FRACTION' starts and ends with a vowel involves intricate calculations of permutations and combinations. By breaking down the problem into manageable parts, we can clearly see the role of each component in the final probability. This concept can be applied to a wide range of similar problems in probability, offering a deeper insight into the fundamental principles of combinatorial mathematics.