UPSC NDA I 2025 Mathematics Paper

The number of selections of r things from 5n things is equal to the number of selections of n+r things from 5n things.
This gives the equation:
(

5n

r

)=(

5n

n+r

)
Using the symmetry property of binomial coefficients, we know that:
(

5n

r

)=(

5n

5n−r

)
Thus, comparing the two sides:
r=5n−(n+r)
Simplifying the equation:
‌r=5n−n−r
‌2r=4n
‌r=2n
∴ The value of r is 2n.