**Introduction**

The number of telephone connections that can be allotted with eight digits from the number 0 to 9 can be calculated using the concept of permutations and combinations. In this case, we have a set of 10 digits (0 to 9) and we need to select 8 digits to form a telephone number. Let's break down the solution into the following steps:

**Step 1: Understanding the Problem**

To find the number of telephone connections that can be allotted with eight digits, we need to determine the total number of possible combinations of 8 digits from the set of 10 digits (0 to 9).

**Step 2: Using Combinations**

In this problem, we need to use combinations instead of permutations. The reason is that the order of the digits does not matter in a telephone number. For example, 12345678 and 87654321 are considered the same telephone number.

**Step 3: Calculating the Number of Combinations**

To calculate the number of combinations, we can use the formula for combination:

nCr = n! / (r! * (n - r)!)

Where n is the total number of digits (10) and r is the number of digits to be selected (8).

Using this formula, we can calculate the number of combinations as follows:

10C8 = 10! / (8! * (10 - 8)!)

**Step 4: Simplifying the Calculation**

To simplify the calculation, we can use the factorial notation. For example, 10! can be written as 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

Using this notation, the calculation becomes:

10C8 = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) * (2 * 1))

**Step 5: Simplifying the Calculation Further**

By simplifying the calculation further, we can cancel out common terms in the numerator and denominator:

10C8 = (10 * 9) / (2 * 1)

**Step 6: Final Calculation**

After simplifying the expression, we get:

10C8 = 45

Therefore, the number of telephone connections that can be allotted with eight digits from the number 0 to 9 is 45.

**Conclusion**

In conclusion, there are 45 possible telephone connections that can be allotted with eight digits from the number 0 to 9. This calculation is based on the concept of combinations, where the order of the digits does not matter.