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Question

The value of $$\displaystyle ^{ 47 }{ { C }_{ 4 } }+\sum _{ r=1 }^{ 5 }{ ^{ 52-r }{ { C }_{ 3 } } } $$ is equal to


A
47C6
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B
52C5
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C
52C4
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D
none of these
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Solution

The correct option is C $$^{ 52 }{ { C }_{ 4 } }$$
We need value of $$={ _{  }^{ 47 }{ C } }_{ 4 }^{  }+\sum _{ r=1 }^{ 5 }{ { _{  }^{ 52-r }{ C } }_{ 3 }^{  } } $$
$$={ _{  }^{ 47 }{ C } }_{ 4 }^{  }+{ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C } }_{ 3 }^{  }+{ _{  }^{ 49 }{ C } }_{ 3 }^{  }+{ _{  }^{ 48 }{ C } }_{ 3 }^{  }+{ _{  }^{ 47 }{ C } }_{ 3 }^{  }$$
$$={ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C } }_{ 3 }^{  }+{ _{  }^{ 49 }{ C } }_{ 3 }^{  }+{ _{  }^{ 48 }{ C } }_{ 3 }^{  }+{ _{  }^{ 47 }{ C } }_{ 3 }^{  }+{ _{  }^{ 47 }{ C } }_{ 4 }={ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C } }_{ 3 }^{  }+{ _{  }^{ 49 }{ C } }_{ 3 }^{  }+{ _{  }^{ 48 }{ C } }_{ 3 }^{  }+{ _{  }^{ 48 }{ C } }_{ 4 }^{  }$$
$$={ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C} }_{ 3 }^{  }+{ _{  }^{ 49 }{ C } }_{ 3 }^{  }+{ _{  }^{ 49 }{ C } }_{ 4 }^{  }={ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C } }_{ 3 }^{  }+{ _{  }^{ 50 }{ C } }_{ 4 }^{  }$$
$$={ _{  }^{ 51 }{ C } }_{ 3 }^{  }+{ _{  }^{ 51 }{ C } }_{ 4 }^{  }={ _{  }^{ 52 }{ C } }_{ 4 }^{  }$$

Mathematics

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