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Question

# The number of non-negative integral solutions of a+b+c+d=n,nϵN, is

A
n+3P2
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B
(n+1)(n+2)(n+3)6
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C
n1Cn4
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D
none of these
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Solution

## The correct option is B (n+1)(n+2)(n+3)6a+b+c+d=nNow nϵN and a,b,c,dϵZ+Consider n=1We geta+b+c+d=1a=0,b=0,c=0,d=1a=0,b=0,c=1,d=0a=0,b=1,c=0,d=0a=1,b=0,c=0,d=0Hence we have 4 solutions=4C3 solutions=1+3C3 solutionsLet n=2we geta=0,b=0,c=0,d=2a=0,b=0,c=1,d=1a=0,b=1,c=0,d=1a=0,b=0,c=2,d=0a=0,b=1,c=1,d=0a=0,b=2,c=0,d=0a=1,b=1,c=0,d=0a=2,b=0,c=0,d=0a=1,b=0,c=0,d=1a=1,b=0,c=1,d=0Hence total 10 solutions=5C3=2+3C3Hence for a+b+c+d=n we will have n+3C3 solutions.=(n+3)(n+2)(n+1)3!=(n+3)(n+2)(n+1)6

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