P(shared birthday ) = 1 – P(no one share a birthday)
$\qquad= 1-\frac{n.(n-1)(n-2)\ldots (n-k+1)}{n^k}$
Here, $n= 365$ days, and we need to find $k$ such that probability exceed $0.1$
$\implies 0.1 \leq 1-\frac{365.364.363\ldots (365-k+1)}{365^k}$
$\implies \frac{365.364.363\ldots (365-k+1)}{365^k} \leq 0.9$
Trying the values, $k = 10.$