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 ( B 10 B

Question

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 ​ ( B 10 ​ B r ​ − C 10 ​ A r ​ ) is equal to

TestHub · Accepted Answer

A r ​ = Coefficient of x r in ​ ( 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 ​ ( B 10 ​ B r ​ − C 10 ​ A r ​ ) = r = 1 ∑ 10 ​ A r ​ B 10 ​ B r ​ − r = 1 ∑ 10 ​ A r ​ C 10 ​ A r ​ = r = 1 ∑ 10 ​ 10 C r ​ 20 C 10 ​ 20 C r ​ − r = 1 ∑ 10 ​ 10 C r ​ 30 C 10 ​ 10 C r ​ = r = 1 ∑ 10 ​ 10 C 10 − r ​ 20 C 10 ​ 20 C r ​ − r = 1 ∑ 10 ​ 10 C 10 − r ​ 30 C 10 ​ 10 C r ​ ​ ​ = 20 C 10 ​ r = 1 ∑ 10 ​ 10 C 10 − r ​ ⋅ 20 C r ​ − 30 C 10 ​ r = 1 ∑ 10 ​ 10 C 10 − r ​ 10 C r ​ = 20 C 10 ​ ( 30 C 10 ​ − 1 ) − 30 C 10 ​ ( 20 C 10 ​ − 1 ) = 30 C 10 ​ − 20 C 10 ​ = C 10 ​ − B 10 ​ ​

Mathematics - Binomial Theorem Question with Solution | TestHub

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