The Probability of Vowels Being Together in the Word “Equation”

Consider the word “equation.” If you were to take the letters of this word and place them at random in a row, what would be the probability that all the vowels are together?

Identify the Vowels and Consonants

The word "equation" consists of the following letters:

Vowels: e, u, a, i, o

Consonants: q, t, n

There are 5 vowels and 3 consonants.

Treat All Vowels as a Single Unit

In order to solve this problem, we need to treat all the vowels (e, u, a, i, o) as a single unit. We can denote this block as V. This unit can be placed in different positions within the word, along with the individual consonants (q, t, n). Thus, we have the following units to arrange:

n - V (the block of vowels)

n - q

n - t

n - n

This gives us a total of 4 units to arrange.

Calculate the Arrangements of Units

The total arrangements of these 4 units can be calculated using the factorial of 4, which is 4!.

4! 4 × 3 × 2 × 1 24

Arrange the Vowels

Inside the vowel block (V), the vowels e, u, a, i, o can be arranged among themselves. The number of arrangements of these 5 vowels can be calculated using the factorial of 5, which is 5!.

5! 5 × 4 × 3 × 2 × 1 120

Total Arrangements with Vowels Together

The total arrangements where all vowels are together can be calculated by multiplying the arrangements of the 4 units by the arrangements of the 5 vowels inside the unit V.

4! × 5! 24 × 120 2880

Calculate Total Arrangements of the Letters

The total number of arrangements of the letters in "equation" can be calculated using the factorial of 8, which is 8!.

8! 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 40320

Probability that All Vowels are Together

The probability P that all vowels are together can be found by dividing the favorable arrangements by the total arrangements.

P frac{4! × 5!}{8!} frac{2880}{40320}

To simplify this fraction:

P frac{1}{14}

Thus, the probability that all the vowels in the word “equation” are together is frac{1}{14}.

Additional Insights

Another way to approach this problem is by considering the arrangement of the vowels and consonants separately. If we consider the vowels as a single unit, we have 4 units to arrange (the vowel unit plus the 3 consonants). The number of ways to arrange these 4 units is 4!.

Within the vowel unit, the 5 vowels can be arranged in 5! ways. Therefore, the total number of arrangements with the vowels together is 4! × 5!.

The probability can again be calculated as:

P frac{4! × 5!}{8!} frac{2880}{40320} frac{1}{14}

Conclusion

By treating the vowels as a single unit and calculating the arrangements accordingly, we can determine the probability that all vowels in the word “equation” are together. The final probability is frac{1}{14}, indicating a relatively low likelihood of this specific arrangement occurring.