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 _ r=1 ¹⁰ A_r (B₁₀ B_r-C₁₀ A_r ) is equal to

Step-by-step solution

A_r= Coefficient of x^r in aligned & (1+x)¹⁰= ¹⁰ C_r & B_r= Coefficient of x^r in & (1+x)²⁰= ²⁰ C_r & C_r= Coefficient of x^r in & (1+x))³⁰= ³⁰ C_r & _ r=1 ¹⁰ A_r (B₁₀ B_r-C₁₀ A_r ) & = _ r=1 ¹⁰ A_r B₁₀ B_r- _ r=1 ¹⁰ A_r C₁₀ A_r & = _ r=1 ¹⁰ ¹⁰ C_r 20 10 ²⁰ C_r & - _ r=1 ¹⁰ ¹⁰ C_r 30 10 ¹⁰ C_r & = _ r=1 ¹⁰ 10 10-r 20 10 ²⁰ C_r & - _ r=1 ¹⁰ 10 10-r 30 10 ¹⁰ C_r & aligned aligned & = 20 10 _ r=1 ¹⁰ 10 10-r ²⁰ C_r & - 30 10 _ r=1 ¹⁰ 10 10-r ¹⁰ C_r & = 20 10 ( 30 10 -1 )- 30 10 ( 20 10 -1 ) & = 30 10 - 20 10 =C₁₀-B₁₀ a