How many mn tuples are there??

Let $ \{x_1, x_2, \ldots , x_n \} $ and $ \{y_1, y_2, \ldots, y_n\} $ be sets of non-negative integers such that $ \sum \limits_{i=1}^n x_i = a $, $ \sum \limits_{i=1}^n y_i = b $, and $x_i + y_i \geq c$ for all $i = 1,2,\ldots, n$.

We are also given that $c \leq \frac{a+b}n$ . $a,b,c$ are all positive integers.

Find the number of ordered solutions of the $2n$ -tuplets of positive integers, $( x_1, x_2, \ldots, x_n, y_1, y_2, \ldots , y_n )$ .

For example, if $n=3,a=4,b=6,c=2$ , then the number of ordered 6-tuplets of positive integers, $(x_1, x_2, x_3, y_1, y_2, y_3)$ is 168.

I wonder if it is possible to count $m\cdot n$ tuples. For example, in this case, $m=2$ because $x_i, y_i$ have $n$ each, so 2 $(=m)$ times $n$ is $2n$ tuples.

#NumberTheory

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$