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Permutations and Combinations question

2009 · Shift 0 · Q43
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Permutations and Combinations question

2009 · Shift 0 · Q43

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :
  1. A
    at least 500 but less than 750
  2. B
    at least 750 but less than 1000
  3. C
    at least 1000
  4. D
    less than 500
View written solutionFree

Correct answer: C

  1. Select the books

We need:

  • 444 novels from 666 different novels
  • 111 dictionary from 333 different dictionaries

Number of ways to choose the novels: (64)=15\binom{6}{4} = 15(46​)=15

Number of ways to choose the dictionary: (31)=3\binom{3}{1} = 3(13​)=3

So, total ways to select the books: 15×3=4515 \times 3 = 4515×3=45

  1. Arrange them with the dictionary always in the middle

There are 555 books in total. If the dictionary must always be in the middle, then its position is fixed at the 3rd3^{\text{rd}}3rd place.

The remaining 444 selected novels can be arranged in the other 444 positions in: 4!=244! = 244!=24 ways.

  1. Total number of arrangements

Hence, total number of required arrangements is: (64)⋅(31)⋅4!\binom{6}{4} \cdot \binom{3}{1} \cdot 4!(46​)⋅(13​)⋅4! =15×3×24= 15 \times 3 \times 24=15×3×24 =1080= 1080=1080

  1. Match with options

Since 1080≥1000,1080 \ge 1000,1080≥1000, it falls in:

Option C: at least 1000

  1. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So, the answer agrees with the stored correct answer.

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