jmc

In general, $$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$$

Consider the decimal part separately, so we're trying to find the greatest integer less than $(10+0.3{)}^3$. By the Binomial Theorem in the above expansion, this is equal to $$10^3+3(10^2)(.3)+3(10)(.3^2)+.3^3$$ Expanding these terms, we have that $(10.3{)}^3=1000+90+2.7+.027$. From this, we find that the greatest integer less than this quantity is $\boxed{1092}$.