I need to find number of combination for word which has following constraint a. The word can have exactly 5 characters b. The word can have digits [0-9], upper case letters [A-Z]and lowercase letters [a-z] c. The word must contain exactly one uppercase letter and exactly one digit characters.
My approach
The word is comprised of 5 characters. For each character can be filled with digit [0-9], uppercase letter [A-Z] or lowercase letter [a-z]. So for the first place we have = 10 +26 +26 = 62 combinations. Similarly we have 62 combinations for each position.
Therefore total number of combinations are=62 * 62 * 62* 62 *62 =916132832
Answer = Total number of combination - combination which does not follow 3rd condition
I am not able to figure out how to calculate number of combinations which does not follow 3rd condition.
$26$ ways to choose the capital letter and $5$ choices for its place.
$10$ ways to choose the digit and $4$ places to put it.
Concerning the remaining three places where to fit small letters, you shall distinguish among:
- three equal letters: $26$ choices, $1$ way to arrange
- two equal letters: $26 \times 25$ choices, $3$ different arrangements
- all different letters: $26 \times 25 \times 24$ ways.
total: $\quad26 \cdot 5 \cdot 10 \cdot 4 \cdot 26 \cdot\left( 1+25 \cdot 3+25 \cdot24 \right)=91 395 200$