Check Your Skills Answer Key

1. 12

   If you use slots and labels, 3 would go in the first slot (maybe labeled "A-B road"), and 4 would go in the second slot (maybe labeled "B-G road"). Now multiply the number of choices for each leg of the trip:
   3 × 4 = 12.
   

2. 18

   Kyle has 3 choices of pants, 3 choices of shirts, and 2 choices involving a tie (yes or no). Label the first slot "P," the second slot "S," and the third slot "T." Put the numbers into the slots. Finally, 
   multiply: 3 × 3 × 2= 18.

3. 120

   This question is asking for the number of ways to order 5 differently colored rings with no restrictions. So compute 5 factorial:

   5! = 5 × 4 × 3 × 2 × 1  = 120

4. 5,040

   The 7 letters in a word with all distinct letters (such as DEPOSIT) are distinct objects. There are 7 slots they can go into: the first position in the word, the second position in the word, and so on. 

   So the letters can be arranged in 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 different ways. (These rearrangements are called anagrams.)

5. 56

   Write down 5 slots for the 5 people Peggy chooses for her team. All 5 slots should be labeled the same:

   _____		_____		_____		_____		_____
   On Team		On Team		On Team		On Team		On Team

   Now fill in the slots. Peggy has 8 friends to choose from for the first slot, then 7 for the second, and on down the line.

   __8__		__7__		__6__		__5__		__4__
   On Team		On Team		On Team		On Team		On Team

   Finally, you multiply those numbers together and divide by 5!, the factorial of the number of repeated labels:

   

6. 700

   For the 7 men, you have 3 identical slots. Here's the computation:

   

   For the 6 women, you have a separate set of 3 identical slots:

   

   Finally, multiply the choices to get the total: 35 × 20 = 700 different ways to form the committee.

   By the way, this is considerably fewer than the number of ways to choose 6 out of 13 people without regard to gender.