How many strings of five uppercase English letters are there
(a) that start or end with the letters BO (in the order), if letters can be repeated? (inclusive or)
(b) that start with an X, if letters can be repeated?
(c) that start with the letters BO (in that order), if letters can be repeated?
(d) if letters can be repeated?
To find the number of strings in each case, we can apply the concept of permutations and combinations. Here is how we can find the answers to each question:
(a) To find the number of strings that start or end with the letters BO, we can consider two cases: strings that start with BO and strings that end with BO.
For strings that start with BO, we have the fixed pattern "BOXXXX" (where "X" represents any uppercase English letter). Since we have 26 choices for each "X," there are 26^4 possible combinations for the remaining four letters. Therefore, there are 26^4 strings that start with BO.
For strings that end with BO, we have the fixed pattern "XXXXBO." Similarly, there are 26^4 possible combinations for the four preceding letters. Therefore, there are also 26^4 strings that end with BO.
To get the inclusive OR between these two cases, we can add the number of strings that start with BO and the number of strings that end with BO. Thus, the answer is 26^4 + 26^4.
(b) To find the number of strings that start with an X, we have a fixed pattern "XYYYYY" (where "Y" represents any uppercase English letter). Similar to the previous case, there are 26^5 possible combinations for the remaining five letters. Therefore, the answer is 26^5.
(c) To find the number of strings that start with the letters BO, we have the fixed pattern "BOXXXXX" (where "X" represents any uppercase English letter). There are 26^5 possible combinations for the five preceding letters. Therefore, the answer is 26^5.
(d) If letters can be repeated, there are 26 choices freely available for each of the five positions (as there are 26 uppercase English letters). Thus, the answer is 26^5.
To summarize:
(a) 26^4 + 26^4
(b) 26^5
(c) 26^5
(d) 26^5