For $r = 0, 1, \ldots, 10$, 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 $\sum_{r=1}^{10} A_r(B_{10}B_r - C_{10}A_r)$ is equal to

For r = 0, 1, …, 10, let Aᵣ, Bᵣ and Cᵣ denote, respectively, the coefficient of xʳ in the expansions of (1+x)¹⁰, (1+x)²⁰ and (1+x)³⁰. Then ∑ᵣ₌₁¹⁰ Aᵣ(B₁₀Bᵣ - C₁₀Aᵣ) is equal to $A_r = \text{coeff of } x^r \text{ in } (1+x)^{10}$ $B_r \text{ in } (1+x)^{20}, \quad C_r \text{ in } (1+x)^{30}$ $\sum_{r=1}^{10} A_r\left(B_{10}B_r - C_{10}A_r\right)$ $A_r = \binom{10}{r}, \quad B_r = \binom{20}{r}, \quad C_r = \binom{30}{r}$ $S = B_{10}\sum_{r=1}^{10} A_r B_r - C_{10}\sum_{r=1}^{10} A_r^{2}$ $\sum_{r=0}^{n}\binom{n}{r}^{2} = \binom{2n}{n}$ $\sum_{r=0}^{n}\binom{n}{r}\binom{m}{r} = \binom{n+m}{n}$ $\sum_{r=0}^{10} A_r^{2} = \sum_{r=0}^{10}\binom{10}{r}^{2} = \binom{20}{10} = B_{10}$ $\sum_{r=1}^{10} A_r^{2} = B_{10} - 1$ $\sum_{r=0}^{10} A_r B_r = \sum_{r=0}^{10}\binom{10}{r}\binom{20}{r} = \binom{30}{10} = C_{10}$ $\sum_{r=1}^{10} A_r B_r = C_{10} - 1$ $S = B_{10}(C_{10} - 1) - C_{10}(B_{10} - 1)$ $S = B_{10}C_{10} - B_{10} - C_{10}B_{10} + C_{10}$ $S = C_{10} - B_{10}$ $\boxed{\sum_{r=1}^{10} A_r\left(B_{10}B_r - C_{10}A_r\right) = C_{10} - B_{10}}$ $\text{Answer: option (4)}$