JEE MAIN - Mathematics (2025 - 2nd April Evening Shift - No. 3)
The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :
_2nd_April_Evening_Shift_en_3_1.png)
5880
840
960
5760
Explanation
_2nd_April_Evening_Shift_en_3_2.png)
Let $x, y, z$ be the number of box which are filled
$\Rightarrow 1 \leq x \leq 3,1 \leq y \leq 3,1 \leq z \leq 2$
| $x$ | $y$ | $z$ | Number of ways |
|---|---|---|---|
| 3 | 1 | 1 | ${ }^3 \mathrm{C}_3 \cdot{ }^3 \mathrm{C}_1 \cdot{ }^2 \mathrm{C}_1=6$ |
| 2 | 2 | 1 | ${ }^3 \mathrm{C}_2 \cdot{ }^3 \mathrm{C}_2 \cdot{ }^2 \mathrm{C}_1=18$ |
| 1 | 3 | 1 | ${ }^3 \mathrm{C}_1 \cdot{ }^3 \mathrm{C}_3 \cdot{ }^2 \mathrm{C}_1=6$ |
| 2 | 1 | 2 | ${ }^3 C_2 \cdot{ }^3 C_1 \cdot{ }^2 C_2=9$ |
| 1 | 2 | 2 | ${ }^3 \mathrm{C}_1 \cdot{ }^3 \mathrm{C}_2 \cdot{ }^2 \mathrm{C}_2=9$ |
Total ways $=(48)$ to fill boxes
Now to arrange $a, b, c, d$ and $e$
Number of ways will be 48.5! = 5760
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