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Mathematics · Permutations and Combinations

JEE Main 2025 — 2 April, Evening Shift — Question 31

The number of ways, in which the letters A,B,C,DA, B, C, D, EE 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:

Question figure
  1. Option A:

    960

  2. Option B:

    5760

    Correct
  3. Option C:

    840

  4. Option D:

    5880

Answer: B

Step-by-step solution

Let x,y,zx, y, z be the number of box which are filled

⇒1≤x≤3,1≤y≤3,1≤z≤2\Rightarrow 1 \leq x \leq 3,1 \leq y \leq 3,1 \leq z \leq 2

xyzNumber of ways
3113C3⋅3C1⋅2C1=1⋅3⋅2=6{}^3C_3 \cdot {}^3C_1 \cdot {}^2C_1 = 1 \cdot 3 \cdot 2 = 6
2213C2⋅3C2⋅2C1=3⋅3⋅2=18{}^3C_2 \cdot {}^3C_2 \cdot {}^2C_1 = 3 \cdot 3 \cdot 2 = 18
1313C1⋅3C3⋅2C1=3⋅1⋅2=6{}^3C_1 \cdot {}^3C_3 \cdot {}^2C_1 = 3 \cdot 1 \cdot 2 = 6
2123C2⋅3C1⋅2C2=3⋅3⋅1=9{}^3C_2 \cdot {}^3C_1 \cdot {}^2C_2 = 3 \cdot 3 \cdot 1 = 9
1223C1⋅3C2⋅2C2=3⋅3⋅1=9{}^3C_1 \cdot {}^3C_2 \cdot {}^2C_2 = 3 \cdot 3 \cdot 1 = 9

Total ways =(48)=(48) to fill boxes

Now to arrange a,b,c,da, b, c, d and ee

Number of ways will be 48.5!=576048.5!=5760

Solution figure

Answer key and solution verified before publishing.

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Exam
JEE Main 2025
Paper
2 April, Evening Shift
Subject
Mathematics
Chapter
Permutations and Combinations
Topic
Applications of Permuations and Combination