The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?

Explanation:

The word given is ZOLOGY. We need to find the probability that the consonants and vowels occur alternatively in all possible arrangements.

• Vowels: O, O, O
• Consonants: Z, L, G, Y

There are 3 vowels and 4 consonants.

The total number of ways the letters of the word "ZOOLOGY" can be arranged (without any restrictions) is:

Formula: Total arrangements = n! / r₁! r₂! ... r_k!

Where:

• n = Total number of letters = 7
• r₁ = Frequency of O = 3

So, the total number of arrangements is:

Total arrangements = 7! / 3! = 5040 / 6 = 840

We need to place the consonants and vowels alternately:

• Consonants will occupy the odd positions (1, 3, 5, 7).
• Vowels will occupy the even positions (2, 4, 6).

The number of ways to arrange the consonants (Z, L, G, Y) in the 4 odd positions is:

4! = 24

The number of ways to arrange the vowels (O, O, O) in the 3 even positions is:

3! / 3! = 1

Thus, the number of favorable arrangements is:

Favorable arrangements = 4! × (3! / 3!) = 24 × 1 = 24

The probability is given by the ratio of favorable arrangements to the total arrangements:

Probability = Favorable arrangements / Total arrangements = 24 / 840 = 1 / 35

The probability that the consonants and vowels occur alternatively is 1 / 35.