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JEE Main & Advanced Sample Paper JEE Main Sample Paper-44

  • question_answer
    The value of\[^{100}{{C}_{0}}{{\,}^{200}}{{C}_{100}}{{-}^{100}}{{C}_{1}}{{\,}^{199}}{{C}_{100}}{{+}^{100}}{{C}_{2}}{{\,}^{198}}{{C}_{100}}\]\[{{-}^{100}}{{C}_{3}}^{197}{{C}_{100}}+...{{+}^{100}}{{C}_{100}}^{100}{{C}_{100}}\] is equal to

    A)  1                                

    B)  -1               

    C)  0                                

    D)  2

    Correct Answer: A

    Solution :

     \[^{100}{{C}_{0}}{{\,}^{200}}{{C}_{100}}{{-}^{100}}{{C}_{1}}{{\,}^{199}}{{C}_{100}}{{+}^{100}}{{C}_{2}}{{\,}^{198}}{{C}_{100}}-\] \[^{100}{{C}_{3}}{{\,}^{197}}{{C}_{100}}+...{{+}^{100}}{{C}_{100}}{{\,}^{100}}{{C}_{100}}\] Coefficient ox \[{{x}^{100}}\] in \[^{100}{{C}_{0}}\,{{(1+x)}^{200}}{{-}^{100}}{{C}_{1}}\,(1+x){{\,}^{199}}\] \[{{+}^{100}}{{C}_{2}}\,{{(1+x)}^{198}}-...+{{\,}^{100}}{{C}_{100}}\,{{(1+x)}^{100}}\] = Coefficient of \[{{x}^{100}}\] in \[{{(1+x)}^{100}}{{[}^{100}}{{C}_{0}}{{(1+x)}^{100}}{{-}^{100}}{{C}_{1}}\,{{(1+x)}^{99}}+\] \[^{100}{{C}_{2}}\,{{(1+x)}^{98}}-...{{+}^{100}}{{C}_{100}}]\] = Coefficient of \[{{x}^{100}}\] in \[{{(1+x)}^{100}}\cdot {{x}^{100}}=1\]


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