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JEE MAIN Aptitude QUESTION #4084
Question 1

A librarian is organizing a special display on a narrow shelf that has exactly 5 slots. She must select books from the following collection:

  • 5 unique Physics books

  • 4 unique Mathematics books

The display must follow these strict rules:

  1. Exactly 3 Physics books and 2 Mathematics books must be chosen.

  2. The Mathematics books must never be placed next to each other.

  3. A specific Physics book (P1) is a "must-have" and must always be included in the selection.

The total number of possible arrangements for this display is:

A)  B)  C)  D) 

  • Less than 400
  • At least 400 but less than 800
  • At least 800 but less than 1200
  • 1200 or more✔️
Correct Answer Explanation

To solve this, you have to break it down into the same "Select then Arrange" flow:

  1. Selection ($C$): * We need 3 Physics books. Since P1 is mandatory, we only need to choose 2 more from the remaining 4. Calculation: $\binom{4}{2} = 6$.

    • We need 2 Math books from 4. Calculation: $\binom{4}{2} = 6$.

    • Total selection combinations: $6 \times 6 = 36$.

  2. Arrangement ($P$): * We have 5 books (3 Physics, 2 Math).

    • The "Gap Method" is best for the constraint "Math books not together."

    • First, arrange the 3 Physics books: $3! = 6$ ways.

    • This creates 4 possible gaps ( _ P _ P _ P _ ).

    • We must place the 2 Math books into these 4 gaps: $^4P_2 = 12$ ways.

    • Total arrangements per selection: $6 \times 12 = 72$.

  3. Final Calculation:

    • $36 \text{ (selections)} \times 72 \text{ (arrangements)} = 2592$.

Correct Answer: D (1200 or more)