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  • 10 identical ball are distributed in 5 different boxes kept in a row and labeled <img alt="\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \text{ and } \mathrm{E}" src="https://entrancecor

10 identical ball are distributed in 5 different boxes kept in a row and labeled \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \text{ and } \mathrm{E}. Find the number of ways in which the ball can be distributed in the boxes if no two adjacent boxes remain empty.
 

Option: 1

789


Option: 2

875
 


Option: 3

771
 


Option: 4

692


Answers (1)

best_answer

Case i : When no box remains empty

Then number of ways of distribution is { }^{10-1} \mathrm{C}_{5-1}={ }^9 \mathrm{C}_4=126

Case ii : Exactly one box is empty 

Number of ways = selection of one box which is empty x distribution of 10 objects in remaining 4 boxes

={ }^5 \mathrm{C}_1 \cdot{ }^9 \mathrm{C}_3=420

Case iii : Exactly two remains empty

Number of ways 

= selection of two boxes which are empty but not consecutive \times distribution of 10 objects in remaining e boxes

\begin{aligned} & =(\text { selection of any two boxes }- \text { two adjacent }){ }^9 \mathrm{C}_2 \\ & =\left({ }^5 \mathrm{C}_2-4\right) \times{ }^9 \mathrm{C}_2 \\ & =6 \times 36=216 \end{aligned}

Case iv : Exactly three empty

There is only 1 way to select three empty boxes if no two adjacent 

Hence number of ways 1 { }^9 C_1=9

Thus total number of ways =126+420+216+9=771

Posted by

Shailly goel

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