How many lists of 100 numbers (1 to 10 only) add to 700? Each number is from one to ten inclusive only.  There are $100$ numbers in the ordered list. The total must be $700$.
How many such lists?
Note: if, as it happens, this is one of those math problems where only an approximation is known, that would be great.  (My guess is it's not "that" big, around $10^{20}$ maybe?)
Thank you!!

So, to be clear say you have a dice with sides labelled $1$-$10$. You roll it $100$ times, once every $10$ seconds in order. The result is, if you will, a specific array of $100$ numbers (each being $1$-$10$), each position labelled $1$ to $100$. The array must then add to $700$; how many such arrays??

Just to be absolutely clear, I believe the total of "all" such lists (so, there is no requirement to add to $700$; it can add to anything), is indeed, simply $1$ googol, ie, $10^{100}$.

In the language of compositions, what you want are the number of $A$-restricted compositions of $n=700$ into $k=100$ parts where $A=\{1,2,\ldots,a\}$ with $a=10$. While I don't know a closed-form for this number, one can express such counting succinctly using formal power series. To represent our set $A$ of allowed terms, we use the formal polynomial $f_{a}(x)=\sum\limits_{j=1}^{a} x^j = \dfrac{x-x^{a+1}}{1-x}$. If we square this, then the coefficient of a term $x^n$ will represent the number of ways in which two integers in $A$ can add to a total of $n$. This gives us the following result:

The number of $A$-restricted compositions, with $A=\{1,2,\ldots a\}$, of $n$ into $k$ parts is the $n$th coefficient of $f_{10}(x)^k$, which we can express as $[x^n]\left(\dfrac{x-x^{a+1}}{1-x}\right)^k.$

Thus in the case at hand what we want to find (or at least estimate) is $\boxed{\displaystyle\left[x^{700}\right]\left(\dfrac{x-x^{11}}{1-x}\right)^{100}}$. Amazingly, this can be pulled off rather easily by WolframAlpha, yielding the rather impressive result of $$
\boxed{
\begin{align}
12113063910758377468145174162592296408571398929 &\\
1260198434849317132762403014516376282321342995 &\approx 1.2 \times 10^{92}
\end{align}
}
$$
For approximations, I suggest perusing Sedgwick and Flajolet's Analytic Combinatorics which is available for download on the book site. I may dig into there myself to see if anything is known or can be estimated about such numbers.
