How many permutations of the letters ABCDEFGH contain the string ABC?
Question
How many permutations of the letters ABCDEFGH contain the string ABC?
Solution 1
To find the number of permutations of the letters ABCDEFGH that contain the string ABC, we can break down the problem into smaller steps:
Step 1: Count the number of ways to arrange the remaining letters (DEFGH) after fixing the string ABC in place.
Since ABC is fixed, we have 5 remaining letters (DEFGH) to arrange. The number of ways to arrange these 5 letters is given by 5!.
Step 2: Count the number of positions where the string ABC can be placed within the arrangement of the remaining letters.
Since ABC is a string of 3 letters, there are 5 possible positions where it can be placed within the arrangement of the remaining 5 letters (DEFGH).
Step 3: Multiply the results from Step 1 and Step 2 to get the total number of permutations.
Multiplying the number of ways to arrange the remaining letters (5!) by the number of positions where the string ABC can be placed (5), we get:
5! * 5 = 120 * 5 = 600
Therefore, there are 600 permutations of the letters ABCDEFGH that contain the string ABC.
Solution 2
To find the number of permutations of the letters ABCDEFGH that contain the string ABC, we can break down the problem into smaller steps:
Step 1: Count the number of ways to arrange the remaining letters (DEFGH) after fixing the string ABC in place.
Since ABC is fixed, we have 5 remaining letters (DEFGH) to arrange. The number of ways to arrange these 5 letters is given by 5!.
Step 2: Count the number of positions where the string ABC can be placed within the arrangement of the remaining letters.
Since ABC is a string of 3 letters, there are 5 possible positions where it can be placed within the arrangement of the remaining 5 letters (DEFGH).
Step 3: Multiply the results from Step 1 and Step 2 to get the total number of permutations.
Multiplying the number of ways to arrange the remaining letters (5!) by the number of positions where the string ABC can be placed (5), we get:
5! * 5 = 120 * 5 = 600
Therefore, there are 600 permutations of the letters ABCDEFGH that contain the string ABC.
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