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Question

A student is to answer $$10$$ out of $$13$$ questions in an examination such that he must choose at least 4 from first five questions. The number of choices available to him is


A
196
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B
280
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C
346
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D
140
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Solution

The correct option is B $$196$$
The number of choices available to him is

$$\displaystyle{  }^{ 5 }{ { C }_{ 9 } }\times {  }^{ 8 }{ { C }_{ 6 } }+{  }^{ 5 }{ { C }_{ 5 } }\times {  }^{ 8 }{ { C }_{ 5 } }$$

$$\displaystyle=\frac { 5! }{ 4! \, 1! } \times \frac { 8! }{ 6!\, 2! } +\frac { 5! }{ 5!\, 6! } \times \frac { 8! }{ 5!\, 3! } $$
$$\displaystyle=5\times\frac { 8\times 7 }{ 2 } +1\times \frac { 8\times 7\times 6 }{ 3\times 2 }$$

$$\displaystyle =5\times 4\times 7+8\times 7$$

$$=140+56$$

$$=196$$

Maths

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