Calculating Arrangements of MANPOWER with P and W Separated by At Least One Letter
Calculating Arrangements of MANPOWER with P and W Separated by At Least One Letter
The word 'MANPOWER' consists of 8 distinct letters. We need to determine the number of possible arrangements where the letters 'P' and 'W' are separated by at least one other letter. Let's break down the problem step by step.
Total Arrangements
The total number of arrangements of the letters in 'MANPOWER' can be calculated using the factorial of the number of letters. Since there are 8 distinct letters:
Total Arrangements
[text{Total arrangements} 8! 40320]
Arrangements with P and W Together
To find the number of arrangements where 'P' and 'W' are together, we can treat 'P' and 'W' as a single entity or block. This reduces the problem to arranging 7 blocks: (PW) M A N O E R. We then calculate the number of arrangements of these 7 blocks and account for the internal arrangements of 'P' and 'W':
Number of Arrangements of Blocks
[text{Arrangements of 7 blocks} 7!]
Since 'P' and 'W' can be arranged internally in 2 ways (PW or WP), we multiply by 2:
Total Arrangements with PW Together
[text{Total arrangements with PW together} 7! times 2 5040 times 2 10080]
Arrangements with P and W Separated
To find the arrangements where 'P' and 'W' are separated by at least one other letter, we subtract the arrangements where they are together from the total number of arrangements:
Calculating Arrangements with P and W Separated
[text{Arrangements with P and W separated} text{Total arrangements} - text{Arrangements with PW together}]
[text{Arrangements with P and W separated} 40320 - 10080 30240]
Summary
Thus, we have:
Total arrangements: 40320 Arrangements with P and W separated: 30240Keyword-Based Calculation
To solve the problem using a different approach, we can consider the arrangement of 'P' and 'W' such that they are treated as one 'letter' and then calculate the rest:
Eliminating P and W Beside Each Other
We need to eliminate the cases where 'P' and 'W' are beside each other. This can be calculated as follows:
2 times 7 times 6!
[2 times 7 times 6! 10080]
This accounts for the two sets (PW and WP) and the number of ways to arrange the remaining 6 letters.
2 times 7!
Alternatively, treating 'PW' or 'WP' as one 'letter', we multiply by 2:
[2 times 7! 10080]
From here, we subtract the cases where 'P' and 'W' are together from the total arrangements to get the final solution:
40320 - 10080 30240
Therefore, the final answer is:
Total arrangements: 40320 Arrangements with P and W separated: 30240By following these steps and using the provided keywords, you can effectively calculate the number of arrangements for any similar problem involving permutations and combinatorics.