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Question

# The number of non negative integral solution of x1+x2+x3+4x4=20 is

A
536
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B
443
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C
655
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D
none of these
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Solution

## The correct option is A 536The number of non negative integral solution of given equation is equal to the coefficient of x20 in (1+x+x2⋯)(1+x+x2⋯)(1+x+x2⋯)(1+x4+x8⋯) =coeff. of x20in(1−x)−1(1−x)−1(1−x)−1(1−x4)−1 =coeff. of x20 in (1−x)−3×(1−x4)−1 =coeff. of x20 in (1+x4+x8+⋯)×(1+ 3C1x+ 4C2x2+ 5C3x3+ 6C4x4+⋯) =1+ 6C4+ 10C8+ 14C12+ 18C16+ 22C20 =536 Alternate Solutions: x1+x2+x3+4x4=20 Now puting x4=0,1,2,3,4,5 we get x1+x2+x3=20,x1+x2+x3=16⋯,x1+x2+x3=4,x1+x2+x3=0 Now for number of solutions of all the eqaution =22C2+18C2+14C2+10C2+6C2+2C2=231+153+91+45+15+1=536

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