how many ways can the letters ALABAMA be arranged: a) If there are no restrictions except indistinguishable letters? b) If the letters B and M must be togheter?
what was the most popular choice?
To calculate the number of ways the letters in the word "ALABAMA" can be arranged, we can use the concept of permutations.
a) If there are no restrictions except indistinguishable letters, then all 7 letters in "ALABAMA" can be arranged. To calculate the number of permutations, we need to find the factorial of the total number of letters:
Total Number of Letters: 7
Number of Ways: 7!
The factorial of a number means multiplying that number by all the positive integers below it. So, in this case, the calculation would be:
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
Therefore, there are 5040 different ways to arrange the letters in "ALABAMA" with no restrictions.
b) If the letters B and M must be together, we can treat these two letters as a single entity. So, we can rearrange "AL,A,B,A,M,A" instead of "ALABAMA." Now we have 5 entities to arrange: "AL," "A," "B," "A," and "M."
To calculate the number of permutations in this case, we need to consider that the "A's" appear twice. Therefore, we would have to divide the total number of permutations by the factorial of the number of repeated letters.
Total Number of Entities: 5
Number of Ways: 5! / (2!)
Here, (2!) represents the factorial of the number of repeated letters, which is the factorial of 2.
Calculating this:
5! / (2!) = (5 x 4 x 3 x 2 x 1) / (2 x 1) = 120 / 2 = 60
Therefore, there are 60 different ways to arrange the letters in "ALABAMA" when the letters B and M are together.