In this set of answers, I have taken the value to an actual integer. You will not need to do this on the exam for these larger numbers because calculators are not allowed, and I don't want to spend time testing if you know how to multiply. I have done it here so that if you used a different method, it should still evaluate to reach the same integer value.

1. Answer the following about the sequence of numbers between 5 and 125 (inclusive):
   1. How many integers are between 5 and 125 (inclusive)? ANSWER: $125-5+1=121$
   2. How many integers are between 5 and 125 (inclusive) which are divisible by 3? ANSWER: $2*3,3*3,...,40*3,41*3=6,9,...,120,123$
      $41-2+2=40$
   3. How many integers are between 5 and 125 (inclusive) which are divisible by 5? ANSWER: $1*5,2*5,...,24*5,25*5 = 5,10,...,120,125$
      $25-1+1=25$
   4. How many integers are between 5 and 125 (inclusive) which are divisible by both 3 and 5? ANSWER: $1*15,2*15, ..., 7*15,8*15 = 15, 30, 105, 120$
      $8-1+1=8$
   5. How many integers are between 5 and 125 (inclusive) which are divisible by either 3 or 5? ANSWER: $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 40+25-8=57$
   6. How many integers are between 5 and 125 (inclusive) which are divisible by neither 3 nor 5? ANSWER: Total number - number that are divisible $= n(U) - n(A \cup B) = 121-57=64$
   7. How many integers are between 5 and 125 (inclusive) which are divisible by 3 but not divisible by 5? ANSWER: number divisible by 3 - number of those that are divisible by 5 $= n(A) - n(A \cap B) = 40 - 8 = 32$
2. For the next set of questions, assume you have a of 6 dogs. Some dogs are big (marked with a b), some are middle sized (marked with a m), and some are small (marked with an s). These dogs are {Alpi (b),Bingo (s), Congo (b), Delfi(s), Elf (s), Fred (m)}. Answer the following questions about the dogs in your kennel.
   1. How many different ways can you select two dogs to take for a walk at the same time? ANSWER: ${6 \choose 2} = \frac{6!}{4!2!} = 15$
   2. How many different ways can you select two dogs to take for a walk at the same time assuming you can't handle 2 big dogs at the same time? ANSWER: Partition and use Addition Rule
      Two non-big dogs + One big and one non-big dog
      ${4 \choose 2} + {2 \choose 1}{4 \choose 1} = 6 + 8 = 14$
   3. How many different ways can you assign the dogs all to leashes (assuming you have 6 leashes in 6 different colors)? ANSWER: Steps and use multiplication rule
      Select the leash for the first dog = 6
      $*$ Select the leash for the second dog = 5
      $*$ ...
      $*$ Select the leash for the last dog = 1
      $= 6!$
   4. How many different ways can you assign the dogs all to leashes (assuming you have 6 leashes in 6 different colors), but also assuming you have only two leashes that can handle the big dogs? ANSWER: Steps and use the multiplication rule
      Select the leashes for the big dogs
      $*$ Select the leashes for the other dogs
      $= 2!*4! = 2*24 = 48$
   5. How many different ways can you divide the dogs up for walking among your 3 volunteers - assuming each volunteer must walk 2 dogs each? ANSWER: Steps and use the multiplication rule
      Select the dogs for the first walker
      $*$ select the dogs for the second walker
      $*$ select the dogs for the third walker
      $={6 \choose 2}{4 \choose 2}{2 \choose 2} = 15*6*1=90$
   6. How many different ways can you line up the dogs (single file) for a dog show? ANSWER: Linear permutation of all 6 dogs $= 6! = 720$
   7. How many different ways can you line up the dogs (single file) for a dog show assuming the large dogs must come first, followed by the mediup then the small dogs? ANSWER: Multiplication rule of the Linear permutations for the individual sizes
      $= 2!*1!*3! = 2 * 1 * 6 = 12$
   8. How many different ways can you create a line in the dog show of four finalists? ANSWER: r-Permutation : $P(6,4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{720}{2} = 360$
   9. How many different ways can you distribute the 15 doggie treats you have (assuming the doggie treats are indistinguishable)? (note: it can be that one dog gets all of the treats - you are not trying to distribute them evenly.) ANSWER: 15 indistinguishable doggie treats into 6 categories
      $\frac{(15+(6-1))!}{15!(6-1)!} = \frac{20!}{15!5!}= 15504$
   10. How many diffent ways can you assign them to the 4 run areas at your kennel? (note: these ``runs'' are distinguishable, but you could be assinging them all to the same run or as evenly as possible.) ANSWER: 6 distinguishable dogs into 4 distinguishable runs
       number of ways for each dog to select one of the four runs
       Dog 1 has 4 runs to select from, dog 2 has 4 runs to select from ..., Dog 6 has 4 runs to select from
       $= 4^6 = 3096$
   11. Assuming you have 6 collars (6 different colors) and 6 leashes (6 different colors), how many ways can you assign the collars and leashes to your six dogs so they can go for a walk? ANSWER: Use the multiplication rule with the two individual steps of assigning collars and then assigning leashes.
       $=6!*6!=720^2=518400$
   12. Assuming you have 5 families interested in adopting dogs, how many different ways can the dogs be given to those families? (note: you are assuming there is no personal preference taken into account.) ANSWER: r-Permutation of assigning 5 of the 6 dogs to distinguishable families
       $= P(6,5) = \frac{6!}{(6-5)!}= \frac{6!}{1!}= 6!=720$
   13. Assuming all of your dogs are male, and the neighbor's dog turns out to be pregnant, how many different ways could your dogs be the father of her puppies? ANSWER: Select the one of your dogs that is the father $= {6 \choose 1} = 6$
       If you want to assume it may be another dog from somewhere else (none of yours are the father this is just one more possibility $= {7 \choose 1} = 7$
   14. Assuming you must take one dog of each size with you, how many diffent ways do you have to select your companions? ANSWER: select the one dog from each set and use the multiplication rule to get the whole
       $= {2 \choose 1}{1 \choose 1}{3 \choose 1} = 2 * 1 * 3= 6$

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