Here are the solutions to your problems: 1. A password consists of 4 letters followed by 3 digits. How many different are possible? Step 1: Determine the number of choices for each position. For letters, there are 26 possible choices (A-Z). For digits, there are 10 possible choices (0-9). Assuming repetition is allowed for both letters and digits. Step 2: Calculate the total number of possibilities for the letters. Number of choices for the 1st letter: 26 Number of choices for the 2nd letter: 26 Number of choices for the 3rd letter: 26 Number of choices for the 4th letter: 26 Total letter combinations = $26 \times 26 \times 26 \times 26 = 26^4$ Step 3: Calculate the total number of possibilities for the digits. Number of choices for the 1st digit: 10 Number of choices for the 2nd digit: 10 Number of choices for the 3rd digit: 10 Total digit combinations = $10 \times 10 \times 10 = 10^3$ Step 4: Multiply the possibilities for letters and digits to get the total number of different passwords. $$ \text{Total passwords} = 26^4 \times 10^3 $$ $$ \text{Total passwords} = 456976 \times 1000 $$ $$ \text{Total passwords} = 456,976,000 $$ The number of different passwords possible is $\boxed{\text{456,976,000}}$. 2. In a group of 7 people, how many ways can a president and a vice president be selected? Step 1: Identify the type of problem. Since the positions (President and Vice President) are distinct, the order in which people are selected matters. This is a permutation problem. Step 2: Define $n$ and $k$. Total number of people ($n$) = 7 Number of positions to fill ($k$) = 2 (President, Vice President) Step 3: Use the permutation formula. The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$. $$ P(7, 2) = \frac{7!}{(7-2)!} $$ $$ P(7, 2) = \frac{7!}{5!} $$ $$ P(7, 2) = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} $$ $$ P(7, 2) = 7 \times 6 $$ $$ P(7, 2) = 42 $$ Alternatively, you can think of it as: Number of choices for President: 7 people After choosing the President, number of choices for Vice President: 6 remaining people Total ways = $7 \times 6 = 42$ The number of ways a president and a vice president can be selected is $\boxed{\text{42}}$.