The number of $$non-negative$$ integral solutions of $$x1$$ $$+$$ $$x2$$ $$+$$ $$x3$$ $$+$$ $$4x4$$ $$=$$ $$20$$ is

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Infinity Learn · Accepted Answer

Number of $$non-negative$$ integral solutions of the given $$equation=$$ coefficient of $$x20$$ $$in(1$$−$$x)$$−$$1(1$$−$$x)$$−$$1(1$$−$$x)$$−$$11$$−$$x4$$−$$1$$.$$=$$ coefficient of $$x20$$ in (1$$−$$x)−$$31$$−$$x4$$−$$1=$$ coefficient of $$x20$$ in $$1+3C1x+4C2x2+5C3x3+6C4x4+$$…$$+10C8x8$$…$$+14C12x12+$$…$$+18C16x16+$$…$$+22C20x20+$$…×$$1+x4+x8+x12+x16+x20+$$…$$=1+6C4+10C8+14C12+18C16+22C20=1+6C2+10C2+14C2+18C2+22C2=1+6.51.2+10.91.2+14.131.2+18.171.2+22.211.2=1+15+45+91+153+231=536$$.

The number of non-negative integral solutions of x₁ + x₂ + x₃ + 4x₄ = 20 is

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The number of non-negative integral solutions of x1 + x2 + x3 + 4x4 = 20 is

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The number of non-negative integral solutions of x₁ + x₂ + x₃ + 4x₄ = 20 is