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Home » A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
CBSE | Class 11 | Combinations | Exercise 17.2 | Maths | RD Sharma
A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?

Given:

Total number of questions

    \[=\text{ }12\]

Total number of questions to be answered

    \[=\text{ }7\]

Number of ways = (No. of ways of answering 5 questions from group 1 and 2 from group 2) + (No. of ways of answering 4 questions from group 1 and 3 from group 2) + (No. of ways of answering 3 questions from group 1 and 4 from group 2) + (No. of ways of answering 2 questions from group 1 and 5 from group 2)

    \[={{(}^{6}}{{C}_{5}}~\times {{~}^{6}}{{C}_{2}})+{{(}^{6}}{{C}_{4}}~\times {{~}^{6}}{{C}_{3}})+{{(}^{6}}{{C}_{3}}~\times {{~}^{6}}{{C}_{4}})+{{(}^{6}}{{C}_{2}}\times {{~}^{6}}{{C}_{5}})\]

By using the formula,

    \[^{n}{{C}_{r}}~=\text{ }n!/r!\left( n\text{ }-\text{ }r \right)!\]

RD Sharma Solutions for Class 11 Maths Chapter 17 – Combinations image - 10

RD Sharma Solutions for Class 11 Maths Chapter 17 – Combinations image - 11

    \[=\text{ }\left( 6\times 15 \right)\text{ }+\text{ }\left( 15\times 20 \right)\text{ }+\text{ }\left( 20\times 15 \right)\text{ }+\text{ }\left( 15\times 6 \right)\]

Or,

    \[=\text{ }90\text{ }+\text{ }300\text{ }+\text{ }300\text{ }+\text{ }90\]

    \[=\text{ }780\]

∴ The total no. of ways of answering

    \[7\text{ }questions\text{ }is\text{ }780\text{ }ways\]

 

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