Answer
$5$ significant digits.
Work Step by Step
See the table below, where we calculate $P(50)=2.56(b)^{61}$
for the year 2011, which is 61 years after 1950.
We start from 12 significant digits for b, and then reduce to 11,10, etc.
We see that up to 5 significant digits, the rounded result is 309, which is what we want.
When we drop to 4 significant digits, the rounded result is $310$.
So, our answer is that we need $5$ significant digits.