Opalg said:
I do not know of any exact way to solve that inequality. But the problem only asks you to "estimate" how large $k$ must be. So you need to have some way of estimating that size of that binomial coefficient. Doing it by trial and error, using a calculator, my estimate is that $k$ must be extremely large (around 450, I think). In any case, the building would have to be a good deal taller than any existing skyscraper.
$450$ is a good estimate. Here's a mathematically nicer way to finish:
a different way of approaching the second half of the problem is to recognize it is equivalent to a Birthday Problem -- what $k_0$ is required to get probability of no 'collisions' bounded below by 0.9? This is amenable to Poisson approximation. But it also can be done directly with analytical bounds -- we have a lower bond via union bound / generalized Bernoulli inequality and an upper bound from the all important inequality $1 +x \leq e^x$ for any $x\in \mathbb R$
The union bound estimate is a lower bound given basically by a triangular number $\lambda = \frac{\binom{10}{2}}{k_0}$
so
$0.9 \leq 1 - \lambda = 1 - \frac{\binom{10}{2}}{k_0} \leq \text{actual probability} \leq \exp\big( -\frac{\binom{10}{2}}{k_0} \big) = \exp\big(-\lambda\big)$
focusing on the lower bound
$0.9 \leq 1 - \frac{\binom{10}{2}}{k_0} $
we can rescale by $k_0$ and simplify to get
$\frac{1}{10} k_0 \geq \binom{10}{2} $
$ k_0 \geq 10\cdot\binom{10}{2} $
which is satisfied with equality for $k_0 = 450$
= = = =
as for the upper bound
$\text{actual probability} \leq \exp\big( -\frac{\binom{10}{2}}{k_0} \big)$
which is $\approx 0.9048$ for $k_0$ = 450.
Also of interest, the upper bound is $\approx 0.89953$ for $k_0 = 425$
This tells you the estimate is sharp.I left computation of $\text{actual probability} $ as an open item for OP. Hopefully it is clear how to compute it and then arrive at the inequalities. If not, I can fill in more details.