In how many ways can 8 different dolls be packed in 5 identical gift boxes such that no box is empty if any of the boxs hold all of the toys?
Answers
Given : 8 different dolls to be packed in 5 identical gift boxes
To Find : Number of ways if no box is empty
Step-by-step explanation:
5 identical gift box
Hence there are 3 ways numbers of dolls allocations are as boxes are identical.
Box Box Box Box Box ( 5 Boxes)
4 1 1 1 1
3 2 1 1 1
2 2 2 1 1
Box1 Box2 Box3 Box 4 Box 5
4 1 1 1 1
Ways = ⁸C₄.⁴C₁.³C₁.²C₁ / 4! = 70 as 4 boxes are identical
Box1 Box2 Box3 Box 4 Box 5
3 2 1 1 1
Ways = ⁸C₃.⁵C₂.³C₁.²C₁/3!= 560 as 3 boxes are identical
Box1 Box2 Box3 Box 4 Box 5
2 2 2 1 1
Ways = ⁸C₂.⁶C₂.⁴C₂.²C₁/3!.2! = 420 ( as 3 boxes and 2 boxes are identical)
Total Ways =70 + 560 + 420
= 1050
There are 1050 ways in which 8 different dolls be packed in 5 identical gift boxes such that no box is empty
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