A password must be 8 characters long and include exactly 3 uppercase letters, 4 lowercase letters, and 1 digits. How many different passwords are possible?Option: 1 420  Option: 2 320  Option: 3 280  Option: 4 200

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 3 out of 8 positions for the uppercase letters. This can be calculated as $\mathrm{C(8,3)}$, which is " 8 choose 3 ":

$\mathrm{ C(8,3)=8 ! /(3 ! \times(8-3) !)=8 ! /(3 ! \times 5 !)=(8 \times 7 \times 6) /(3 \times 2 \times 1)=56 . }$

Next, let's consider the placement of lowercase letters. We need to choose 4 out of the remaining 5 positions for the lowercase letters. This can be calculated as $\mathrm{C(5,4)}$, which is " 5 choose 4 ":

$\mathrm{ C(5,4)=5 ! /(4 ! \times(5-4) !)=5 ! /(4 ! \times 1 !)=5 . }$

Similarly, for the placement of the digit, we need to choose 1 out of the remaining 1 position for the digit. This can be

calculated as $\mathrm{C(1,1)}$, which is " 1 choose 1 ":

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

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

$\mathrm{ 56 \times 5 \times 1=280 \text {. } }$

Therefore, there are 280 different passwords possible that are 8 characters long, include exactly 3 uppercase letters, 4 lowercase letters, and 1 digit.