Show that the number of solutions in nonneg. int. of the ineq. $$x_1+x_2+\cdots +x_n\leq M,$$ where $M$ is a nonneg. int., is $C(M+n,n)$.
2026-04-11 22:10:01.1775945401
Cardinality of Solutions to an Inequality
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HINT: The number of solutions in non-negative integers to
$$x_1+x_2+\ldots+x_n\le M$$
is the same as the number of solutions in non-negative integers to
$$x_1+x_2+\ldots+x_n+x_{n+1}=M\;,$$
which is a standard stars-and-bars problem.