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How many solutions are there to the inequality x1 + x2 + x3 ≤ 11 where x1, x2, x3 are nonnegative integers? Hint: introduce a variable x4 such that x1 + x2 + x3 + x4 = 11.

Ans. C(4+11-1,11).

My doubt is if the question had been x1 + x2 + x3 >= 11. then how would have we solved it ?

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We have to solve $x_1 + x_2 + x_3 \geq 11$, under $\mathbb{N}$.

Notice that if either of the variable is greater than or equal to 11, the other two variables don't matter anymore - the statement has become true. So no matter what value you put to the other two variables, the statement will remain true.

Hence, we have infinite many solutions.

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