Ten identical balls are distributed in 5 different boxes kept in a row and labeled `A, B, C, D and E.` The number of ways in which the ball can be dis

Monjur Alahi

Ten identical balls are distributed in 5 different boxes kept in a row and labeled `A, B, C, D and E.` The number of ways in which the ball can be distributed in the boxes if no two adjacent boxes remains empty
A. `789`
B. `875`
C. `771`
D. `692`


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Salim Ahmed Kawsar

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Correct Answer - C
`(c )` Case I : When no box remains empty
Then number of ways of distribution is `.^(10-1)C_(5-1)=^(9)C_(4)=126`
Case ii : Exactly one box is empty
Number of ways `=` selection of one box which is empty `xx` distribution of `10` objects in remaining `4` boxes
`=^(5)C_(1)*^(9)C_(3)=420`
Case iii : Exactly two remains empty
Number of ways
`=` selection of two boxes which are empty but not consecutive `xx` distribution of `10` objects in remianing e boxes
`=("selection of any two boxes"-"two adjacent") ^(9)C_(2)`
`=(^(5)C_(2)-4)xx^(9)C_(2)`
`=6xx36=216`
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`

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