UPSC NDA I 2015 Mathematics Paper

Concept:Permutations with repeated items are calculated using the formula $\frac{n!}{p!q!...}$, where $p,q,...$ are counts of repeated letters.To handle "never together" conditions, subtract the count of arrangements where the items are together from the total count of arrangements.Explanation:The word 'NATION' has 6 letters: N appears twice; the other letters (A, I, T, O) are distinct.Total distinct arrangements of all letters = $\frac{6!}{2!} = \frac{720}{2} = 360$.The vowels are A, I, O — treat them as a single block.Now we have 4 items: the vowel-block, N, N, and T.Since the two N's are identical, the number of arrangements of these 4 items = $\frac{4!}{2!} = \frac{24}{2} = 12$.Inside the block, the 3 vowels can be arranged in $3! = 6$ ways.Thus, total arrangements where all three vowels are together = $12 \times 6 = 72$.Therefore, arrangements where the three vowels are never together = total arrangements − arrangements with vowels together = $360 - 72 = 288$.Answer:288 (which corresponds to option C).