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 \sum_{r = 1}^{10} A_{r} \textrm{ } \left(\right. B_{10} \textrm{ }\textrm{ } B_{r} \textrm{ }\textrm{ } - \textrm{ }\textrm{ } C_{10} \textrm{ }\textrm{ } A

Engineering

Mathematics

Properties of Binomial Coefficients

Question

For r = 0, 1, .... 10, let A_r, B_r and C_r denote, respectively, the coefficient of xr 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

B₁₀ – C₁₀

C₁₀ – B₁₀

$A_{10} (B_{10}^{2} - C_{10} A_{10})$

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${\sum_{r = 1}^{10}}^{10} C_{r} (^{20} C_{10}^{20} C_{r} -^{30} C_{10}^{10} C_{r}) = \sum_{r = 1}^{10}^{20} C_{10} (^{10} C_{r}^{20} C_{20 - r}) -^{30} C_{10} \sum_{r = 1}^{10} {(^{10} C_{r})}^{2}$

= ²⁰C¹⁰(¹⁰C²⁰ – 1) – ³⁰C₁₀(²⁰C₁₀ – 1)

= B_10 10 – B₁₀ – B₁₀C10 + C₁₀ = C₁₀ – B₁₀