14.7. a) In how many ways can a natural number $n$ be represented as the sum of $m$ non-negative integers, if the representations $n=x_{1}+\ldots+x_{m}$ and $n=y_{1}+\ldots+y_{m}$ are considered the same if and only if $x_{1}=y_{1}, \ldots, x_{m}=y_{m} ?$

b) The same question for representation as the sum of natural numbers.
14.7. a) Answer: $C_{n+m-1}^{n}$ ways. We can associate the decomposition $n=x_{1}+\ldots+x_{m}$ with a sequence of zeros and ones, where initially there are $x_{1}$ ones, followed by one zero, then $x_{2}$ ones, followed by one zero, and so on; at the end, there are $x_{m}$ ones. In total, this sequence contains $n+m-1$ digits, among which there are $n$ ones. Therefore, we need to choose $n$ items from $n+m-1$ numbered (i.e., distinct) items.

b) Answer: $C_{n-1}^{n-m}$. Each decomposition $n=x_{1}+\ldots+x_{m}$, where $x_{1}, \ldots, x_{m}$ are natural numbers, corresponds to the decomposition $n-m=y_{1}+\ldots+y_{m}$, where $y_{1}=x_{1}-1, \ldots, y_{m}=x_{m}-1$.