Question 11
Find the largest integer that is less than
$$\frac{3^{10}-2^{10}}{10!}\left(\frac1{1!9!2}+\frac1{2!8!2^2}+\frac1{3!7!2^3}+\cdots+\frac1{9!1!2^9}\right)^{-1}.$$
Here, $n!=n\cdot(n-1)\cdots3\cdot2\cdot1$. For example, $5!=5\cdot4\cdot3\cdot2\cdot1=120$.
$$\frac{3^{10}-2^{10}}{10!}\left(\frac1{1!9!2}+\frac1{2!8!2^2}+\frac1{3!7!2^3}+\cdots+\frac1{9!1!2^9}\right)^{-1}.$$
Here, $n!=n\cdot(n-1)\cdots3\cdot2\cdot1$. For example, $5!=5\cdot4\cdot3\cdot2\cdot1=120$.