perfect square

[man111]

The number of integers [tex]n\geq 10[/tex] such that the product [tex]\displaystyle \binom{n}{10}\cdot \binom{n+1}{10}[/tex] is a perfect square.

[pal702004]

$(n+1)(n-9)=x^2$ in positive integers

$(n-4)^2-25=x^2$

$(n-4-x)(n-4+x)=25$

$n-4-x=1$ and $n-4+x=25$

Solution: $n=17,x=12$