UPSC NDA 2 2017 Mathematics Solved Paper

Concept:Permutations: Number of ways to arrange $n$ distinct objects in $r$ specific places is ${}^{n}P_{r} = \frac{n!}{(n-r)!}$.Explanation:Step 1: Total 16 people and 16 chairs — 8 on each side.Four specific men must sit on one particular side (say side A) and two on the other side (side B).Step 2: On side A (8 chairs), choose 4 chairs and arrange the 4 men: ways = ${}^{8}P_{4} = \frac{8!}{4!}$.Step 3: On side B (8 chairs), choose 2 chairs and arrange the 2 men: ways = ${}^{8}P_{2} = \frac{8!}{6!}$.Step 4: After placing these 6 specific men, 10 people remain and exactly 10 chairs are left (since $8+8-4-2=10$).These 10 people can be arranged in the 10 chairs in $10!$ ways.Step 5: Total arrangements = ${}^{8}P_{4} \times {}^{8}P_{2} \times 10!$ = $\frac{8!}{4!} \times \frac{8!}{6!} \times 10!$ = $8! \times 8! \times \frac{10!}{4!\,6!}$.Since $\frac{10!}{4!\,6!} = {}^{10}C_{4} = 210$, the total becomes $210 \times 8! \times 8!$.Answer:$210 \times 8! \times 8!$ (Option C).