UPSC NDA II 2024 Mathematics Paper

Concept:The probability is the ratio of the number of favorable arrangements where consonants and vowels appear alternately to the total number of possible arrangements of the letters in "ZOOLOGY".Explanation:The word "ZOOLOGY" has 7 letters: Z, O, O, L, O, G, Y.Vowels: O, O, O (3 identical vowels).Consonants: Z, L, G, Y (4 distinct consonants).Total number of arrangements without any restriction:$\frac{7!}{3!} = \frac{5040}{6} = 840$.For alternating pattern, the only possible sequence is consonant–vowel–consonant–vowel–consonant–vowel–consonant.Consonants occupy odd positions (1, 3, 5, 7).Vowels occupy even positions (2, 4, 6).Number of ways to arrange the 4 consonants in the 4 odd positions: $4! = 24$.Number of ways to arrange the 3 identical vowels in the 3 even positions: $\frac{3!}{3!} = 1$.Favorable arrangements: $24 \times 1 = 24$.Probability = $\frac{\text{Favorable}}{\text{Total}} = \frac{24}{840} = \frac{1}{35}$.Answer:$\frac{1}{35}$