To solve the problem of finding the number of solutions that satisfy:

$a + b + c + d \leq 10 \quad \text{with} \quad a, b, c, d \geq 1,$

we'll transform the inequality and then apply the combinatorial "stars and bars" method.

Step 1: Substitute Variables

Since each variable $a, b, c,$ and $d$ must be at least 1, we can let: $a' = a - 1, \quad b' = b - 1, \quad c' = c - 1, \quad d' = d - 1,$ where $a', b', c',$ and $d' \geq 0$.

Now the inequality becomes: $(a' + 1) + (b' + 1) + (c' + 1) + (d' + 1) \leq 10.$

Simplifying, we get: $a' + b' + c' + d' \leq 6.$

Step 2: Convert to an Equality with a New Variable

Let $e = 6 - (a' + b' + c' + d')$, where $e \geq 0$. Then, the equation becomes: $a' + b' + c' + d' + e = 6.$

Now, we need to find the number of non-negative integer solutions to this equation, which is a classic combinatorial problem.

Step 3: Apply the Stars and Bars Theorem

The number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = n$ in non-negative integers is given by: $\binom{n + k - 1}{k - 1},$ where $n = 6$ and $k = 5$ (since we have $a', b', c', d',$ and $e$).

Thus, the number of solutions is: $\binom{6 + 5 - 1}{5 - 1} = \binom{10}{4}.$

Step 4: Calculate $\binom{10}{4}$

$\binom{10}{4} = \frac{10!}{4!(10 - 4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210.$

Conclusion

The number of solutions that satisfy $a + b + c + d \leq 10$ with $a, b, c, d \geq 1$ is:

$\boxed{210}.$

Would you like more details on any part of this solution?

---

Here are five related questions that expand on this concept:

1. How would the solution change if the inequality were $a + b + c + d = 10$?
2. How does the "stars and bars" method generalize to more variables?
3. What would happen if we imposed a stricter upper bound on each variable individually?
4. How can we apply similar methods to find integer solutions within a range for each variable?
5. How would the solution change if we allowed one of the variables to be zero?

Tip: The stars and bars method is a powerful technique for counting integer solutions to equations, especially useful when dealing with combinatorial inequalities.