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

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NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. 6/35
  2. 3/35
  3. 2/35
  4. 1/35

Answer (Detailed Solution Below)

Option 4 : 1/35
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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! ... rk!

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.

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