UPSC NDA I 2023 Mathematics Solved Paper

Concept:The binomial coefficient $C(n,r)$ is maximum when $r$ is the integer closest to $\frac{n}{2}$.For even $n$, the maximum is unique at $r = \frac{n}{2}$, i.e., $n = 2r$.For odd $n$, there are two equal maxima at $r = \frac{n-1}{2}$ and $r = \frac{n+1}{2}$, i.e., $n = 2r-1$ and $n = 2r+1$.Explanation:Statement 1: $C(n,r)$ is greatest if $n = 2r$.When $n$ is even, the middle term is unique: $C(n, n/2)$ is the largest.Here $n = 2r$ implies $r = n/2$, so statement 1 is correct.Statement 2: $C(n,r)$ is greatest if $n = 2r-1$ and $n = 2r+1$.When $n$ is odd, the two middle terms $C(n, (n-1)/2)$ and $C(n, (n+1)/2)$ are equal and largest.These correspond to $n = 2r-1$ (since $r = (n+1)/2$) and $n = 2r+1$ (since $r = (n-1)/2$).Thus statement 2 is also correct.Answer:Both statements 1 and 2 are correct. Hence the correct option is C (Both 1 and 2).