The call letter of a radio station must have 4 letters.The first letter must be a K or a W. How many different station call setters can be made if repetitions are not allowed? If repetitions are allowed? I have the answers just need to know the work
number= 2*25*24*23 if repetition are not allowed.
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To find the number of different station call letters that can be made, we can think of it as a permutation problem.
1. If repetitions are not allowed:
When repetitions are not allowed, the number of different combinations can be evaluated using the formula for permutations without repetition.
Since the first letter must be either K or W, there are 2 choices for the first letter.
For the remaining 3 letters, we have 26 options for each letter since repetitions are not allowed.
So the total number of different station call letters can be calculated as:
2 (choices for the first letter) * 26 (choices for the second letter) * 25 (choices for the third letter) * 24 (choices for the fourth letter)
= 2 * 26 * 25 * 24
= 31,200 different station call letters.
2. If repetitions are allowed:
When repetitions are allowed, the number of different combinations can be evaluated using the formula for permutations with repetition.
Again, since the first letter must be either K or W, there are 2 choices for the first letter.
For the remaining 3 letters, we still have 26 options for each letter since repetitions are allowed.
So the total number of different station call letters can be calculated as:
2 (choices for the first letter) * 26 (choices for the second letter) * 26 (choices for the third letter) * 26 (choices for the fourth letter)
= 2 * 26 * 26 * 26
= 35,152 different station call letters.
Therefore, if repetitions are not allowed, there are 31,200 different station call letters, and if repetitions are allowed, there are 35,152 different station call letters.
To find the number of different station call letters that can be made with 4 letters, we need to consider two cases: one where repetitions are not allowed, and another where repetitions are allowed.
Case 1: No repetitions allowed
In this case, we can choose the first letter from the set {K, W}, which gives us 2 choices. For the remaining three letters, we have a total of 24 options (26 letters of the alphabet minus the already chosen first letter) for each position. Therefore, the total number of call letters without repetitions can be determined using the multiplication principle: 2 choices for the first letter multiplied by 24 choices for each of the remaining three letters.
Total number of call letters without repetitions = 2 * 24 * 24 * 24 = 2 * 24^3 = 2 * 13,824 = 27,648.
Case 2: Repetitions allowed
In this case, we can again choose the first letter from the set {K, W}, which gives us 2 choices. However, now we have 26 options (the entire alphabet) for each of the remaining three letters. So, using the multiplication principle, we get:
Total number of call letters with repetitions = 2 * 26 * 26 * 26 = 2 * 26^3 = 2 * 17,576 = 35,152.
Therefore, the number of different station call letters that can be made are:
Without repetitions: 27,648
With repetitions: 35,152.