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

Step-by-step solution

Case-1 : When no box remains empty it is equivalently distributing 10 coins in 5 beggar 00000 ₅ ₄= ^9 C ₄=126 Case-2: Exactly one is empty ^5 C ₁ 000000 ₆ =5 ^9 C ₃=420 Case-3: Exactly two remains empty aligned & ( ^5 C ₂ - two adjacent )^9 C ₂ 0000000 ₇ & (10-4) ^9 C ₂ & 6 36=216 aligned Case-4: Exactly three empty. There is only 1 way to select 3 if no two adjacent Hence 1 ^9 C ₁=9 00000000 Total =771 ways