The main purpose of this question is to find the number of strings of four specific lowercase letters that have the letter $x$ in them.
Bit strings depict subsets of sets, in which a $1$ indicates that the associated component of a set is a part of the subset and a $0$ indicates that it is not included. We frequently need to quantify the number of sequences having the length $k$ that fulfill specific characteristics and label these kinds of sequences as correct. Assume the characteristics controlling these sequences result in the succeeding selection rule for establishing a correct sequence character by character. Suppose a process can be split up into two tasks, with $n_1$ ways to complete the first and $n_2$ ways to complete the second task. Then there are $n_1\cdot n_2$ different approaches for carrying out the process.
To calculate the total number of results for two or more consecutive events, take the product of the number of results for every event simultaneously. For example, if it is required to find the number of potential results when rolling a die and tossing a coin, the product rule can be utilized. It is vital to remember that the occurrences would have to be independent, which means that neither of them affects the other.
It is a fact that there are $26$ letters in the English alphabet.
To obtain the strings of length four, it is required to utilize the product rule. The first event refers to choosing the first bit, the second event refers to choosing the second, the third event refers to choosing the third, and the fourth event refers to choosing the fourth bit. Because of that, we have:
$26\cdot 26 \cdot 26 \cdot 26=26^4=456,976$
To obtain the strings of length four without $x$, it is again required to utilize the product rule. The first event refers to choosing the first bit, the second event refers to choosing the second, the third event refers to choosing the third, and the fourth event refers to choosing the fourth bit. Because of that, we have:
$25\cdot 25 \cdot 25 \cdot 25=25^4=390,625$
Finally, for the strings of length four with at least one $x$ is:
Find the number of bit strings of length $6$.
Because every one of the $6$ bits can be either a $0$ or a $1$, therefore:
$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=2^6=64$