We need to find the minimum value of $(a + b) \times (c + d)$, where $a$, $b$, $c$, and $d$ are single-digit non-zero distinct numbers.

Step 1: Analyze the Expression

The expression $(a + b) \times (c + d)$ depends on the sums $a + b$ and $c + d$. To minimize the product, we should minimize these sums.

Step 2: Minimize the Sums

Since $a$, $b$, $c$, and $d$ are distinct single-digit non-zero numbers, the smallest values for $a$ and $b$ would be 1 and 2, and the smallest values for $c$ and $d$ would be 3 and 4.

So, we calculate: $(a + b) = 1 + 2 = 3$ $(c + d) = 3 + 4 = 7$

Step 3: Calculate the Product

Now, calculate the product: $(a + b) \times (c + d) = 3 \times 7 = 21$

Conclusion

The minimum value of $(a + b) \times (c + d)$ is 21.

Would you like further details or have any questions?

Related Questions:

1. What would the maximum value of $(a + b) \times (c + d)$ be under the same conditions?
2. How would the problem change if the digits were not distinct?
3. What if one of the digits was allowed to be zero?
4. How many different values can $(a + b) \times (c + d)$ take?
5. How does the use of distinct numbers impact the calculation?

Tip:

When minimizing or maximizing expressions involving sums and products, always start by considering the smallest or largest values possible within the given constraints.