Homework 8 -- Combinatorics

1. There are five roads from town A to town B, two roads from town B to town C, and three roads from town A to town C that avoid town B. All the roads are two way streets.
   1. How many ways are there to go from A to C by way of B?
   2. How many ways are there to go from A to C?
   3. In how many ways can we get from A to C and back?
   4. In how many ways can we get from A to C and back if B is passes through at least one.
   
   
2. How many permutations are there of the letters in the following words:
   1. completeness
   2. aardvark
   3. peppy
   
   
3. Evaluate the following:
   1. 6!
   2. P(7,3)
   3. P(5,3)
   4. C(5,2)
   
   
4. There are eight horses in a race. How many win-place-show finishes are possible?

5. Simplify each of the following:
   1. 12!/9!
   2. n!/(n-2)!
   
   
6. How many licence plate numbers are there consisting of three letters followed by three digits?

7. How many 5-card poker hands can be formed from a deck of 52 cards?

8. I plan to buy 3 shirts from a group of 20 and 2 jackets from a rack of 8. How many combinations are possible?

9. How many "words" (strings) are there consisting of nine distinct letters AND using the vowels a,e,i,o,u,y. (Hint: first select the non-vowels, then order)

10. A full house in poker consists of three of one kind and a pair of another. How many different ways are there to get a full house?

11. There are 12 girls and 10 boys in a class. A committee of five is selected for a project. How many possible committees are there containing:
    1. exactly two boys
    2. at least one girl
    
    
12. How many 4-element subsets are there of a 25-element set?