UPSC NDA I and II 2020 Math Solved Paper

Concept:The problem involves arranging letters with a given constraint on positions. We use the multiplication principle and factorial counting to determine the number of valid arrangements.Explanation:The word "ABLE" has 4 letters: two vowels (A, E) and two consonants (B, L). The positions are 1, 2, 3, 4. The even places are positions 2 and 4 – there are 2 even places. Step 1: Choose which 2 of the 2 even places will be occupied by the vowels. The number of ways to choose 2 places out of 2 is ${}^2C_2 = 1$. Step 2: Arrange the 2 vowels in those 2 chosen places. The vowels can be arranged among themselves in $2! = 2$ ways. Step 3: The remaining 2 positions (the odd places) are to be filled by the 2 consonants. The consonants can be arranged in those 2 places in $2! = 2$ ways. Step 4: Multiply the counts from each step: $1 \times 2 \times 2 = 4$ total arrangements. Therefore, the letters of "ABLE" can be arranged so that vowels occupy even places in 4 ways.Answer:Option B – 4 ways.