How many ways different sets of values can be chosen for the $x_s$ , if $x_1 + x_2 + x_3 = 20$?

Question

Your statistics teacher announces a twenty-page reading assignment on Monday that is to be finished by Thursday morning. You intend to read the first $x_1$ pages Monday, the next $x_2$ pages Tuesday, and the final $x_3$ pages Wednesday, where $x_1+x_2+x_3=20$, and each $x_i\geq1$.

In how many ways can you complete the assignment? That is, how many different sets of values can be chosen for $x_1$, $x_2$, and $x_3$?

Attempt: I am learning $nC_k=\frac{n!}{k!(n-k)!}$. Then it seems there are $3$ distinct number of pages, namely: $x_1$, $x_2$ and $x_3$. And number of ways to form combinations of size $20$. I don't understand. I have not learned about other ways, so I have to use $nC_k$.

Answer 1

I overlooked the condition that we're supposed to read a page per day. In that case, we only need $19$ squares:

$$\boxed{\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square}$$

Now we write down that we're going to read $4,6,10$ as follows:

$$\boxed{\square\square\square\blacksquare\square\square\square\square\square\blacksquare\square\square\square\square\square\square\square\square\square}$$

We're taking it for granted that we're going to read one page, and now the number of white boxes count how many extra pages we intend to read.

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OLD ANSWER: (still worth keeping, as it solves the case where we are allowed to read nothing on some days)

Mark 22 spaces on a piece of paper: $$\boxed{\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square\square}$$

Say that we read for $4$ pages, $6$ pages, then $10$ pages. We can represent that by blocking out two places:

$$\boxed{\square\square\square\square\blacksquare\square\square\square\square\square\square\blacksquare\square\square\square\square\square\square\square\square\square\square}$$

We can interpret this as something like, "Read four pages, then go to sleep, then read six pages, then go to sleep, then read ten pages."

So how many ways are there of coloring in these little black boxes?