Answer:
Option C will have greatest number of permutations.
Explanation:
A.
The number of license plates with 4 letters followed by 3 numbers, where letters and digits cannot be repeated.
We have total 26 letters (A_Z) and the letters can't be repeated. So we have 26 letter options for 1st letter
25 letters options for 2nd letter
24 letter options for 3rd letter
23 letter options for 4th letter.
So total options for 4 letters 26*25*24*23 = 358800
Now, 3 numbers are required. We have total 10 numbers (0-9)
10 number options for 1st number
9 number options for 2nd number
8 number options for 3rd number
Total options for 3 numbers= 10*9*8 = 720
total options for 4 letters and 3 numbers will be: 358800 * 720 = 258,336,000
B.
The number of license plates with 3 letters followed by 4 numbers, where letters and digits cannot be repeated.
We have total 26 letters (A_Z) and the letters can't be repeated. So we have 26 letter options for 1st letter
25 letters options for 2nd letter
24 letter options for 3rd letter
So total options for 3 letters= 26*25*24 = 15600
Now, 4 numbers are required. We have total 10 numbers (0-9)
10 number options for 1st number
9 number options for 2nd number
8 number options for 3rd number
7 number options for 4th number
Total options for 4 numbers = 10*9*8*7 = 5040
total options for 4 letters and 3 numbers will be: 15600 * 5040 = 78,624,000
C. The number of license plates with 4 letters followed by 3 numbers, where letters and digits can be repeated.
Now, the letters and digits can be repeated. So we have 26 options for each 4 letters and 10 options for each 3 numbers
26*26*26*26 *10*10*10 = 456,976 * 10,00 = 456,976,000
total options for 4 letters and 3 numbers will be: 456,976,000
D.
The number of license plates with 3 letters followed by 4 numbers, where letters and digits can be repeated.
Now, the letters and digits can be repeated. So we have 26 options for each 3 letters and 10 options for each 4 numbers
26*26*26 *10*10*10*10 = 17,576 * 10,000 = 175,760,000
total options for 3 letters and 4 numbers will be: 175,760,000
So, Option C will have greatest number of permutations.