1.A password consists of 4 letters followed by 3 digits. How many different are possible? 2. In a group of 7 people, how many ways can a president and a vice president be selected?

• 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.

• 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 = $26 \times 26 \times 26 \times 26 = 26^4$

• 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 = $10 \times 10 \times 10 = 10^3$

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?

Since the positions (President and Vice President) are distinct, the order in which people are selected matters. This is a permutation problem.

• Total number of people ($n$) = 7
• Number of positions to fill ($k$) = 2 (President, Vice President)

The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$.

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 $\boxed{\text{42}}$.