Understanding the Letter Arrangement Problem

This problem asks us to find the probability that when the letters of the word ZOOLOGY are arranged randomly, the consonants and vowels appear in an alternating pattern. To solve this, we need to determine the total possible arrangements and the number of arrangements where consonants and vowels alternate.

Analyzing the Word ZOOLOGY

First, let's break down the word ZOOLOGY:

• Total number of letters: 7
• Vowels: O, O, O (3 vowels)
• Consonants: Z, L, G, Y (4 consonants)

Notice that the letter 'O' is repeated 3 times.

Calculating Total Possible Arrangements

The total number of ways to arrange the letters in ZOOLOGY is calculated using the formula for permutations with repetitions.

Total Arrangements = \(\frac{\text{Total Letters}!}{\text{Repetitions}!}\)

In this case, it is:

Total Arrangements = \(\frac{7!}{3!}\)

Calculating the factorials:

• \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)
• \(3! = 3 \times 2 \times 1 = 6\)

So, the total number of arrangements is:

Total Arrangements = \(\frac{5040}{6} = 840\)

Calculating Favorable Arrangements (Alternating Pattern)

For the consonants and vowels to alternate, the arrangement must follow a specific pattern. Since there are 4 consonants and 3 vowels, the only possible alternating pattern is one that starts and ends with a consonant:

Pattern: C V C V C V C

Where 'C' represents a consonant and 'V' represents a vowel.

Arranging the Consonants

There are 4 distinct consonants (Z, L, G, Y) to be placed in the 4 consonant positions (1st, 3rd, 5th, 7th). The number of ways to arrange these 4 distinct consonants is \(4!\).

Ways to arrange consonants = \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Arranging the Vowels

There are 3 vowels (O, O, O) to be placed in the 3 vowel positions (2nd, 4th, 6th). Since all the vowels are identical ('O'), there is only one way to arrange them among themselves (the formula is \(\frac{3!}{3!}\)).

Ways to arrange vowels = \(\frac{3!}{3!} = 1\)

Total Favorable Arrangements

To find the total number of arrangements where consonants and vowels alternate, we multiply the number of ways to arrange the consonants by the number of ways to arrange the vowels.

Favorable Arrangements = (Ways to arrange consonants) \(\times\) (Ways to arrange vowels)

Favorable Arrangements = \(24 \times 1 = 24\)

Calculating the Probability

The probability of an event is calculated as:

Probability = \(\frac{\text{Number of Favorable Arrangements}}{\text{Total Number of Arrangements}}\)

Plugging in our calculated values:

Probability = \(\frac{24}{840}\)

Now, we simplify the fraction:

\(\frac{24}{840} = \frac{12}{420} = \frac{6}{210} = \frac{3}{105} = \frac{1}{35}\)

Therefore, the probability that the consonants and vowels occur alternatively is \(\frac{1}{35}\).