If `ngt3, then ((n-1),(1))(a-1)^(2)-((n-1),(2))(a-2)^(2)+((n-1),(3))(a-3)^(2)-...+(-1)^(n-2)((n-1),(n-1))(a-n+1)^(2)=k` then number of positvie divisors of k, (given 'a' is a prime number)

Answer

C

Explanation

`t_(r)=(-1)^(r-1).^(n-1)C_(r)(a-r)^(2)`
` implies S=underset(r=1)overset(n-1)Sigma(-1)^(r-1).^(n-1)C_(r)(a^(2)-2ar+r^(2))`
`=a^(2)underset(r=1)overset(n-1)Sigma(-1)^(r-1).^(n-1)C_(r)-2aunderset(r=1)overset(n-1)Sigma(-1)^(r-1)r.^(n-1)C_(r)`
`+underset(r=1)overset(n-1)Sigma(-1)^(r-1)r^(2).^(n-1)C_(r)`
`=a^(2)(1)-2a(0)+(0)=a^(2)`