JEE Main 2024 — 29 January, Shift 2 — Question 2

Case 1: 3 shelves empty

$$(8,0,0,0) \;\longrightarrow\; 1 \text{ way}$$

Case 2: 2 shelves empty

$$(7,1,0,0),\; (6,2,0,0),\; (5,3,0,0),\; (4,4,0,0)$$ $$\longrightarrow\; 4 \text{ ways}$$

Case31 shelf empty

$$(6,1,1,0),\; (5,2,1,0),\; (4,3,1,0),\; (4,2,2,0),\; (3,3,2,0)$$ $$\longrightarrow\; 5 \text{ ways}$$

Case 4: 0 shelf empty

$$(5,1,1,1),\; (4,2,1,1),\; (3,3,1,1),\; (2,2,2,2)$$ $$\longrightarrow\; 5 \text{ ways}$$

Total number of ways $= 1 + 4 + 5 + 5 = \boxed{15}$