Question

How many 7 digit telephone numbers can be formed from the digits \(0,1,2, \ldots, 9\), where each telephone number begins with digit \(2 ?\) (a) \(10^{6}\) (b) \(6^{10}\) (c) 101 (d) \({ }^{10} P_{6}\)

Short answer

Answer: (a) \(10^{6}\)

Step-by-step solution

Identifying the first digit

Since each telephone number must begin with the digit 2, the first digit is fixed, and we don't need to consider any other possibilities for it.

Counting possibilities for the remaining digits

Now we need to determine the number of possible choices for the remaining 6 digits. For each position (from the second to the seventh), we have 10 options (0 to 9 inclusive).

Calculating the number of telephone numbers

To find the total number of telephone numbers that can be formed, we multiply the number of possibilities for each of the 6 positions. Since there are 10 options for each position, the multiplication will give us the result: $$ 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^{6} $$

Identifying the correct answer

Comparing the result with the given answer choices, we find that the correct answer is (a) \(10^{6}\).

Key concepts

Combinatorics and Permutations

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is the foundation for assessing the number of possible arrangements (or permutations) of a set of objects, where the order of arrangement is important.

When it comes to telephone numbers, for example, the arrangement of the digits is crucial because the same digits in a different order can represent a completely different telephone number. The term permutations refers to all the possible ways in which a set of things can be ordered or arranged.

Considering our example, with the first digit fixed as '2', we only have to find permutations for the remaining six digits. These can be any digits from 0 to 9, meaning for each position, there are 10 possible digits. Since the positions of the remaining six digits are distinct and can be filled independently of each other, there are 10 to the power of 6, or 1,000,000 unique permutations possible for a 7-digit telephone number starting with '2'.

Probability Fundamentals

Probability is the measure of the likelihood that an event will occur. It quantifies the chance of a particular outcome in an uncertain process, and it's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In the study of telephone numbers, if we were asked whether a randomly chosen 7-digit telephone number starts with a '2', we would utilize probability. Assuming each digit from 0 to 9 has an equal likelihood of being chosen, the probability that a number starts with '2' is \( \frac{1}{10} \), since there is one favorable outcome ('2') out of ten possibilities.

For more complex situations involving combinations and permutations, probability calculations can include factorials and binomial coefficients. However, for this exercise, since we're not looking at the likelihood of forming certain patterns of digits but merely counting the total possibilities, probability concepts such as expected value or variance are not directly applied.

Number Theory Insights

Number theory is a field of mathematics concerned with the properties and relationships of numbers, particularly the integers. It includes the study of prime numbers, modular arithmetic, and other structures that deal with discrete mathematics.

In our example of creating telephone numbers, we are using base-10, which is the everyday number system we are familiar with. The exercise doesn't call for deep number theory concepts, but it's still crucial to understand that each digit in a base-10 number can take an integer value from 0 to 9, producing a total of ten different possibilities for each position.

Fixed and Variable Elements in Number Sequence

In certain situations, like our telephone number, one or more digits might be fixed, which reduces the variability and thus the total count of potential numbers. But for a full 7-digit number with no restrictions, there would be 10 million (\(10^7\)) possibilities. Learning and applying principles of number theory can help students navigate and solve such problems with confidence.