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InterviewSolution
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The coefficient of `x^50` in the polynomial `(x + ^50C_0)(x +3.^5C_1) (x +5.^5C_2).....(x + (2n + 1) ^5C_50)`, isA. `50.2^(50)`B. `50.2^(51)`C. `51.2^(50)`D. `50 .(2^(50) +1)` |
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Answer» Correct Answer - c `(x + ""^(50)C_(0)) (x +3. ""^(50)C_(1))(x + 5 . ""^(50)C_(2))...(x + (2b +1)""^(50)C_(50))` = `x^(51) + x^(50) ""^(50){""^(50)C_(0) + 3 . ""^(50)C_(2)+…+ 101""^(50)C_(50)} + …` `therefore `Coefficient of `x^(50)` = `""^(50)C_(0) +3. ""^(50)C_(1) +5. ""^(50)C_(2)+...+101""^(50)C_(50)` =` sum _(r=0)^(50) (2r +1) ""^(50)C_(r)` `= 2 sum_(r=0)^(50) r .""^(50)C_(r) + sum_(r=0)^(50) ""^(50)C_(r)` `= 2xx 50sum_(r=1)^(50) ""^(49)C_(r-1) + sum_(r=0)^(50) ""^(50)C_(r)` `100 xx2^(49) + 2^(50) = 51.2^(50)` . |
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