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Combinatory

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How many ways can we distribute at most 10 identical balls to 3 boxes?
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10 identical balls in 3 boxes. (assuming boxes to be identical because if boxes were distinct then it should have been mentioned)

$\Rightarrow Part(10,1) + Part(10,2) + Part (10,3)$

$Part(10,1) = 1 $

$Part(n,k) = part(n-1,k-1) + part(n-k,k) $

$Part(10,2) = Part (9,1) + Part(8,2)$  

$Part(8,2) = Part (8,1) + Part (6,2)$

$ Part(6,2) = Part(6,1) + Part (4,2)$

$ Part(4,2) = Part(4,1) + Part (2,2)$

$ Part(2,2) = Part(2,1) + Part (0,2)$

$ Part(2,2) = 1$

$ Part(4,2) = 2$

$ Part(6,2) = 3$

$ Part(8,2) = 4$

$ Part(10,2) = 5$

similarly after calculation $ Part(10,3) = 8$

$\Rightarrow 1 + 5 +8 = 14$

if boxes are distinct,

$\binom{10+3-1}{3-1} = \binom{12}{2} = 66$ 

 

 

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