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In general, find the number of k-tuples  of nonnegative integers such that $$0 \le a_1 \le a_2 \le a_3 \le \dots \le a_k \le n.$$

Feb 29, 2020

$$\text{clearly all the a's must be such that 0 \leq a_k \leq n}\\ \text{given any set of k integers there is exactly one unlabelled ordering of them satisfying above}\\ \text{so we can choose k integers 0\leq a_k \leq n and sort them, and label them accordingly}\\ \text{there are N=(n+1)^k different sets of k integers in the range [0,n]}$$