10.4. Combined Use of Sum and Product Rule

Try these first

1. 1.

   How many ways can 3 students be chosen from a set of 5?

   Hint

   Use C(n, k) = n! / (k!(n-k)!), where n is the total number and k is the number of choices.

2. 2.

   If each student can pick one of 4 different books, how many choices are there for 3 students?

   Hint

   Use the Product Rule: 4^3.

3. 3.

   How many ways can 2 employees be assigned to 3 different offices if they cannot share an office?

   • 3
   • 6
   • 9

   Hint

   Recall the Product Rule.

4. 4.

   Is it true that if you have 15 items in 10 boxes, at least one box must contain more than one item?

   • True
   • False

   Hint

   Think about how distribution must work.

5. 5.

   In a network of 7 friends, where each pair is either friends or enemies, determine the minimum number of people needed to guarantee 3 mutual relationships.

   Hint

   Visualize relationships as graphs.

6. 6.

   For valid passwords of lengths 6, 7, and 8 using specified character sets, solve for total valid combinations accounting for at least one digit.

   Hint

   Group by length for efficient counting.