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 : [2025]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
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 :

[2025]

1 5880

2 5760

3 840

4 960

Ans.
(2)

We have 5 letters A, B, C, D, E.

The following table shows the numbers of ways to fill 5 boxes so that no row remains empty.

Row-I Row-II Row-III Number of ways

3 1 1 ${}^{3}C_{3} \cdot {}^{3}C_{1} \cdot {}^{2}C_{1} = 6$

2 2 1 ${}^{3}C_{2} \cdot {}^{3}C_{2} \cdot {}^{2}C_{1} = 18$

1 3 1 ${}^{3}C_{1} \cdot {}^{3}C_{3} \cdot {}^{2}C_{1} = 6$

2 1 2 ${}^{3}C_{2} \cdot {}^{3}C_{1} \cdot {}^{2}C_{2} = 9$

1 2 2 ${}^{3}C_{1} \cdot {}^{3}C_{2} \cdot {}^{2}C_{2} = 9$

Number of ways to fill boxes = 6 + 18 + 6 + 9 + 9 = 48

Now, these 5 letters can be arranged in 5! ways

$∴$ Total number of ways $48 \times 5! = 5760$ .

Want Us to Email you About Special Offers & Updates?