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At its core, combinatorics is a branch of mathematics focused on counting, arranging, and combining objects in certain orders or sets. This field is crucial for understanding the principles behind how many bit strings consist of a certain number of digits, as in the textbook exercise we’ve explored.

For example, when calculating bit strings of a certain length, combinatorics tells us we're looking into the number of ways we can arrange binary digits (0s and 1s). A string of length one has two possibilities because a binary digit can be either 0 or 1. For a string of length two, there are four possible combinations: 00, 01, 10, and 11. This pattern continues as the string increases in length. The key insight from combinatorics in this scenario is the way we calculate the total possibilities for each string length and then sum them to find the total number of distinct bit strings.

Binary digits, also known as bits, are the fundamental units of digital data and can have a value of either 0 or 1. In the textbook exercise, we use bits to create bit strings, which are sequences of these binary digits.

In a computing context, bits serve as the smallest building block of data, representing a binary choice or state. This simplicity allows for binary systems to be stable and reliable, which is why they are the foundation of modern computing. When we consider bit strings of varying lengths, we start to see the implications of this binary system on the number of different sequences that can be formed, leading us into the concept of permutations.

Permutations involve arranging a set of items in various orders where the arrangement is important. In a bit string context, each bit has two options, and as we create longer strings, we calculate permutations to identify all the possible combinations.

For instance, a 3-bit string has 2 options for each place, leading to a total of 2 x 2 x 2 permutations. Basically, for each additional binary digit we add to a string, we are effectively doubling the number of possible permutations. It’s this principle that allows us to swiftly calculate the number of bit strings for any given length and is a perfect illustration of the exponential growth inherent in such sequences.

Exponential growth occurs when the increase in a quantity is proportional to the current amount. This kind of growth is present in many natural phenomena and is critical for understanding the increase in the number of bit strings as their length grows.

In our bit string scenario, each additional digit results in an exponential increase of the possible combinations. The number of bit strings of a certain length is calculated by raising 2 (the number of possibilities for each digit) to the power of the string length. For instance, the exponential growth of bit strings with each additional digit is clear when you see that a 3-digit string has 8 possibilities (2^3) while a 4-digit string has 16 (2^4). Exponential growth helps highlight the substantial increase in complexity and quantity, even with the addition of a single binary digit to a string.