Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have
exactly one B.
How do I solve this? Please leave an explanation to solve this.
there are 25 letters besides the B. So, each of the other 7 positions has 25 choices, making a total of 8*25^7 ways to form words with one B somewhere.
So what does length 8 mean? So is length like a position?
no - a word of 8 letters
it is 8 letters long
its length is 8
The B can be in any of 8 places; that's why the 8*25^7
To solve this problem, we can use the concept of permutations and combinations.
First, we need to determine the number of positions that the letter 'B' can occupy in the word. Since we want exactly one 'B' in the word, there are 8 possible positions for the 'B' (one in each position).
Next, we need to fill the remaining empty positions with the other letters from the alphabet. Since we are allowed to repeat letters, each of the remaining 7 positions can be filled with any of the 25 letters (excluding 'B').
To find the total number of words, we need to find the number of ways to fill the empty positions with the other letters. Since there are 7 positions, and each position has 25 options, the total number of words would be 25^7.
Finally, we multiply the number of positions for the 'B' (8) with the number of possible combinations for the other positions (25^7) to get the total number of words.
Therefore, the number of words of length 8 that have exactly one 'B' is:
8 * 25^7.