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  • A password must be 14 characters long and include exactly 5 uppercase letters, 5 lowercase letters, and 4 digits. How many different passwords are possible? &nb

A password must be 14 characters long and include exactly 5 uppercase letters, 5 lowercase letters, and 4 digits. How many different passwords are possible?

 

Option: 1

625,456

 


Option: 2

150,968

 


Option: 3

105,875

 


Option: 4

252,252


Answers (1)

To calculate the number of different passwords that satisfy the given criteria, we can use combinations and multiplication principles.

First, let's consider the placement of uppercase letters. We need to choose 5 out of 14 positions for the uppercase letters. This can be calculated as \mathrm{C(14,5)}, which is "14 choose 5 ":

\mathrm{ C(14,5)=14 ! /(5 ! \times(14-5) !)=14 ! /(5 ! \times 9 !)=(14 \times 13 \times 12 \times 11 \times 10) /(5 \times 4 \times 3 \times 2 \times 1)=2002 \text {. } }

Next, let's consider the placement of lowercase letters. We need to choose 5 out of the remaining 9 positions for the lowercase letters. This can be calculated as \mathrm{C(9,5)}, which is "9 choose 5":

\mathrm{ C(9,5)=9 ! /(5 ! \times(9-5) !)=9 ! /(5 ! \times 4 !)=(9 \times 8 \times 7 \times 6) /(4 \times 3 \times 2 \times 1)=126 . }

Similarly, for the placement of digits, we need to choose 4 out of the remaining 4 positions for the digits. This can be calculated as \mathrm{C(4,4)}, which is " 4 choose 4 ":

\mathrm{ C(4,4)=4 ! /(4 ! \times(4-4) !)=4 ! /(4 ! \times 0 !)=1 }

To find the total number of different passwords, we multiply the number of choices for each category:

\mathrm{ 2002 \times 126 \times 1=252,252 }

Therefore, there are 252,252 different passwords possible that are 14 characters long, include exactly 5 uppercase letters, 5 lowercase letters, and 4 digits.

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Kshitij

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