if \$a×b×c<180\$ , then how many possible positive integer solution are there for the inequality?

As most of you know there is a classical question in elementary combinatorics such that if $$a×b×c$$=180 , then how many possible positive integer solution are there for the equation ?

The solution is easy such that $$180=2^2×3^2×5^1$$ and so , for $$a=2^{x_1}×3^{y_1}×5^{z_1}$$ , $$b=2^{x_2}×3^{y_2}×5^{z_2}$$ , $$c=2^{x_3}×3^{y_3}×5^{z_3}$$ .

Then: $$x_1+x_2+x_3=2$$ where $$x_i≥0$$ , and $$y_1+y_2+y_3=2$$ where $$y_i≥0$$ and $$z_1+z_2+z_3=1$$ where $$z_i≥0$$.

So , $$C(4,2)×C(4,2)×C(3,1)=108$$.

Everything is clear up to now.However , i thought that how can i find that possible positive integer solutions when the equation is $$a×b×c<180$$ instead of $$a×b×c=180$$

After , i started to think about it. Firstly , i thought that if i can calculute the possible solutions for $$x_1+x_2+x_3<2$$ where $$x_i≥0$$ , and $$y_1+y_2+y_3<2$$ where $$y_i≥0$$ and $$z_1+z_2+z_3<1$$ where $$z_i≥0$$ , then i can find the solution.However , there is a problem such that when i calculate the solution , i do not include the prime numbers and their multiplicites which is in $$180$$.

For example , my solution does not contain $$1×1×179<180$$

My question is that how can we solve these types of question . Is there any TRICK for include all possible ways ? Moreover ,this question can be generalized for $$a×b×c≤180$$ , then what would happen for it ?