Verified Solution • 2025–26 Session moderate

QJEEM24Ma53a9: JEE Main 2024 Mathematics - Class 11 JEEM24Ma53a9 Solution: Number of ways of arranging 8 identical books into 4 identical…

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

Multiple Choice Options

  • A 18
  • B 16
  • C 12
  • D 15

Step-by-Step Verified Solution

3 Shelf empty: (8,0,0,0)→1 way 3 \text { Shelf empty: }(8,0,0,0) \rightarrow 1 \text { way }

2 shelf empty: (7,1,0,0)(6,2,0,0)(5,3,0,0)(4,4,0,0)]→4 ways \left.2 \text { shelf empty: } \begin{array}{c}(7,1,0,0) \\(6,2,0,0) \\(5,3,0,0) \\(4,4,0,0)\end{array}\right] \rightarrow 4 \text { ways }

1 shelf empty: (6,1,1,0)amp;(3,3,2,0)(4,2,1,0)amp;(4,2,2,0)(4,3,0)amp;]→5 ways \left.1 \text { shelf empty: } \begin{array}{cc}(6,1,1,0) & (3,3,2,0) \\(4,2,1,0) & (4,2,2,0) \\(4,3,0) &\end{array}\right] \rightarrow 5 \text { ways }

0 Shelf empty : (1,2,3,2)amp;(5,1,1,1)(2,2,2,2)(3,3,2,1,1)(4,2,1,1)amp;]→5 ways \left.0 \text { Shelf empty : } \begin{array}{ll}(1,2,3,2) & (5,1,1,1) \\(2,2,2,2) \\(3,3,2,1,1) \\(4,2,1,1) &\end{array}\right] \rightarrow 5 \text { ways }

Total =15=15 ways

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