JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Critical Thinking Questions

  • question_answer If \[x,\ y\] and \[r\] are positive integers, then \[^{x}{{C}_{r}}{{+}^{x}}{{C}_{r-1}}^{y}{{C}_{1}}{{+}^{x}}{{C}_{r-2}}^{y}{{C}_{2}}+.......{{+}^{y}}{{C}_{r}}=\] [Karnataka CET 1993; RPET 2001]

    A) \[\frac{x\ !\ y\ !}{r\ !}\]

    B) \[\frac{(x+y)\ !}{r\ !}\]

    C) \[^{x+y}{{C}_{r}}\]

    D) \[^{xy}{{C}_{r}}\]

    Correct Answer: C

    Solution :

    The result is trivially true for\[r=1,\ 2\]. It can be easily proved by the principle of mathematical induction that the result is true for \[r\] also.

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