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To solve the problem of finding the number of non-negative integer solutions to the equation (x_1 + x_2 + ... + x_n = k) where each (x_i \leq m), we use the stars and bars method combined with the inclusion-exclusion principle. Here's the step-by-step solution:

1. Base Case (No Upper Limit)

Without the upper bound (x_i \leq m), the number of non-negative integer solutions is given by the stars and bars formula:
[ \binom{k + n -1}{n-1} ]
where (\binom{a}) denotes the combination of (a) items taken (b) at a time.

2. Inclusion-Exclusion Principle for Upper Limits

We need to exclude solutions where at least one (x_i > m). Let (A_i) be the set of solutions where (x_i > m).

For a set (A_i):

• If (x_i > m), let (y_i = x_i - (m+1)) (so (y_i \geq 0)).
• The equation becomes (yi + \sum{j \neq i} x_j = k - (m+1)).
• The number of solutions for (A_i) is (\binom{(k - (m+1)) + n -1}{n-1}) if (k \geq m+1), else 0.

For overlapping sets (A{i1} \cap ... \cap A{it}) (t variables exceed (m)):

• Let (y{ik} = x{ik} - (m+1)) for each (ik).
• The equation becomes (\sum{ik} y{ik} + \sum_{j \notin {i1,...,it}} x_j = k - t(m+1)).
• The number of solutions is (\binom{(k - t(m+1)) + n -1}{n-1}) if (k \geq t(m+1)), else 0.

3. Final Formula

Using inclusion-exclusion, the number of valid solutions is:
[ \sum_{t=0}^s (-1)^t \binom{n}{t} \binom{k - t(m+1) + n -1}{n-1} ]

Where:

• (s = \min\left(n, \left\lfloor \frac{k}{m+1} \right\rfloor\right)) (maximum number of variables that can exceed (m)).
• Terms where (k - t(m+1) <0) are zero and can be ignored.

Example

For (n=3, m=2, k=5):

• Total solutions without limits: (\binom{5+3-1}{3-1} = \binom{7}{2}=21).
• Exclude solutions where one (x_i>2): (\binom{3}{1} \binom{5-3+3-1}{3-1} =3 \times \binom{4}{2}=18).
• Overlapping solutions (two variables >2): (\binom{3}{2} \times 0=0) (since (5 <2(3))).

Valid solutions: (21 - 18 =3) (matches the permutations of ((2,2,1))).

Answer:
The number of solutions is:
[ \boxed{\sum_{t=0}^{\min\left(n, \left\lfloor \frac{k}{m+1} \right\rfloor\right)} (-1)^t \binom{n}{t} \binom{k - t(m+1) + n -1}{n-1}} ]

Or, if you need a concrete expression (depending on values of (n,k,m)), substitute the variables into the sum. For example, if (k \leq m), the sum reduces to (\binom{k+n-1}{n-1}).

For general cases, this sum is the standard result for bounded stars and bars.

(\boxed{\sum_{t=0}^s (-1)^t \binom{n}{t} \binom{k - t(m+1) + n -1}{n-1}}) (with (s) as defined above).

If you want to write it concisely (assuming the sum is computed correctly), this is the final answer.

(\boxed{\sum_{t=0}^{\min\left(n, \left\lfloor \frac{k}{m+1} \right\rfloor\right)} (-1)^t \binom{n}{t} \binom{k - t(m+1) + n -1}{n-1}})

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