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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Definition of combinations, Conditional combinations, Division into groups, Derangements

  • question_answer
    \[^{n}{{C}_{r}}{{+}^{n-1}}{{C}_{r}}+......{{+}^{r}}{{C}_{r}}\] = [AMU 2002]

    A) \[^{n+1}{{C}_{r}}\]

    B) \[^{n+1}{{C}_{r+1}}\]

    C) \[^{n+2}{{C}_{r}}\]

    D) \[{{2}^{n}}\]

    Correct Answer: B

    Solution :

    \[^{r}{{C}_{r}}{{+}^{r+1}}{{C}_{r}}{{+}^{r+2}}{{C}_{r}}......{{+}^{n-1}}{{C}_{r}}{{+}^{n}}{{C}_{r}}\] \[{{=}^{r+1}}{{C}_{r+1}}{{+}^{r+1}}{{C}_{r}}{{+}^{r+2}}{{C}_{r}}+.....{{+}^{n-1}}{{C}_{r}}{{+}^{n}}{{C}_{r}}\] \[{{=}^{r+2}}{{C}_{r+1}}{{+}^{r+2}}{{C}_{r}}+.....{{+}^{n-1}}{{C}_{r}}{{+}^{n}}{{C}_{r}}\] \[{{=}^{r+3}}{{C}_{r+1}}+......{{+}^{n-1}}{{C}_{r}}{{+}^{n}}{{C}_{r}}\]. On solving similar way, we get \[^{n-1}{{C}_{r+1}}{{+}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r}}{{=}^{n}}{{C}_{r+1}}{{+}^{n}}{{C}_{r}}{{=}^{n+1}}{{C}_{r+1}}\].


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