# Deep Binomials

**Discrete Mathematics**Level 3

For \( r = 0, 1, ..., 10,\) let \(A_{r}\), \(B_{r}\) and \(C_{r}\) denote, respectively, the coefficient of \(x^{r}\) in the expansions of \((1+x)^{10}\), \((1+x)^{20}\) and \((1+x)^{30}\). Then \(\sum\limits_{r=1}^n A_{r}(B_{10}B_{r} - C_{10}A_{r}) \) is equal to

**Note**: Take \(n = 10\).