Q.

Number of integral solutions to the equation  x+y+z=21, where x1,y3,z4 is equal to_______

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answer is 105.

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Detailed Solution

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x+y+z=21(1)x1,y3,z4x10,y30,z40
Let α=x1,β=y3,γ=z4 so that α0,β0,γ0
(1) becomes α+1+β+3+γ+4=21
               α+β+γ=13
The number of integral solutions of equation is  n+r1cr1=13+31C31=15c2
=15×142×1=15×7=105

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