Which situation will have the greatest number of permutations? a. the number of license plates with 4 letters followed by 3 numbers, where letters and digits cannot be repeated. b. the number of license plates with 3 letters followed by 4 numbers, where letters and digits cannot be repeated. c. the number of license plates with 4 letters followed by 3 numbers, where letters and digits can be repeated. d. the number of license plates with 3 letters followed by 4 numbers, where letters and digits can be repeated. select the best answer from the choices provided a b c d

Option C will have greatest number of permutations.

Step-by-step 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.

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