Question

Find the number a,, of words in letters A, B, C, D, E of length n in which no letter B appears anywhere to the right of any letter A.

Answer 1

The number of words in letters A, B, C, D, E of length n in which no letter B appears anywhere to the right of any letter A is
`\(a_n = 2^n\).`

Step-by-step explanation:

Consider a word of length n formed by the letters A, B, C, D, E. The presence of the letter B to the right of A is restricted. To construct such a word, at each position i (1 to n), we have two choices: either place the letter A or any other letter (C, D, or E). For each position, there are 2 options, and since there are n positions, the total number of words is
`\(2^n\).`

This can be understood by constructing the word from left to right. At the first position, we can either place A or any of the other letters, giving us 2 choices. For the second position, we again have 2 choices for each of the options chosen in the first position, resulting in a total of
`\(2 * 2 = 2^2\)` possibilities. This pattern continues for all n positions, leading to a total of
`\(2^n\)` words.

Therefore, the final answer is
`\(a_n = 2^n\),` indicating the number of valid words of length n with the specified conditions.