Why Are These Discrete Math Problems So Challenging?

[Townsend]

I don't know why but these types of problems always seem to kick my butt. Any how here they are and my best guesses as to the correct answer.

1. Danny has 12 different lures in his tackle box that he takes on a five day fishing trip. On each day of the trip he fishes with the same lure all day then puts the lure back into his tackle box. How many ways can this happen?

12^5 = 248832 ways.

2. Danny has 10 different lures in his big tackle box. He takes 7 of these and puts them in his little tackle box and goes fishing with his friend Brian. Danny takes three of his lures and loans them to Brian, who did not bring any lures. Danny and Brian each choose a lure and begin fishing., How many ways can this happen?

$${10 \choose 7} {7 \choose 3} {4 \choose 1} {3 \choose 1} = 50400$$

3. Danny starts the day with 9 different lures in his tackle box. He fished with one lure until he loses it on a snag. He then puts on a new lure and fishes with it until he loses that lure. He does this until he has lost 5 lures. Then Danny gets sad because he lost so many lures and goes home. How many ways can this happen?

9*8*7*6*5=15,120 ways.

4. Danny takes 4 different lures with him in his little tackle box. He catches 8 different fish that day. Danny tries to remember which fish bit on which lure. How many ways could this have happened?

I think this is equivalent to finding the number of functions from an 8 element set to a 4 element set. So I think the answer is 4^8 = 65536.

5. Danny takes 5 different lures with him in his little tackle box. He catches 13 yellow perch that day. Danny tries to remember how many perch he caught on each lure. How many ways could this have happened?
$${17 \choose 13} = 2,380$$

6. Danny goes to a fishing show. At a lure booth a salesman has 4 identical Rapala lures, 5 identical Mister Twister lures, 8 identical Hula popper lures and 14 identical silver minnow lures. Danny buys 6 lures from the salesman. How many ways could this have happened?

I really have no clue about this one but to take a shot in the dark I would think this would be the same as picking 6 from the total.
$${31 \choose 6}$$

7. Danny has 12 different lures in his big tackle box. He picks out 4 lures and puts them in his little tackle box and tells Brian to pick out three lures to put in his little tackle box. At the end of the day they put all the lures back in Danny's big tackle box. The next day Danny picks out 5 lures to put in his little tackle box and Brian picks out 4 lures to put in his little tackle box. How many ways could this have happened?

Just another shot in the dark but I guess,
$${12 \choose 4} {8 \choose 3} {12 \choose 5} {7 \choose 4} = 768398400$$

8. Danny is going to order lures from a catalog. The catalog has 12 different lures in it. Danny orders a total of 5 lures, how many ways can this happen?
$${16 \choose 5} =4368$$

Sorry if there are too many questions for you to want to tackle (no pun intended) but even if you only look at a couple I would appreciate it.

Thanks

Jeremy

 

[anathema]

Hello Jeremy,

I can definitely understand how these types of problems can be challenging. But don't worry, let's see if we can tackle (pun intended) some of these together.

1. Your answer of 12^5 = 248832 ways is correct. This is because on each day, Danny can choose from 12 different lures, and he does this for 5 days.

2. Your answer of 50400 is also correct. This is because Danny chooses 7 lures from his big tackle box, then Brian chooses 3 lures from the remaining 7, and finally Danny and Brian each choose 1 lure from the remaining 4.

3. Your answer of 15120 is also correct. This is because Danny fishes with 1 lure for the first day, then 1 less lure for each subsequent day until he has lost 5 lures.

4. Your answer of 65536 is correct. This is because for each of the 8 fish, Danny can choose from 4 different lures.

5. Your answer of 2380 is also correct. This is because Danny caught a total of 17 perch, and he could have caught 13 of them on one lure and 4 on another, or any other combination that adds up to 17.

6. Your answer of {31 \choose 6} is correct. This is because Danny is choosing 6 lures from the total of 31 lures that the salesman has.

7. Your answer of 768398400 is correct. This is because Danny first chooses 4 lures from his big tackle box, then Brian chooses 3 lures from the remaining 8, then Danny chooses 5 lures from the remaining 12, and finally Brian chooses 4 lures from the remaining 7.

8. Your answer of {16 \choose 5} = 4368 is correct. This is because Danny is ordering 5 lures from the catalog that has 12 different lures, so he has a total of 16 choices.

I hope this helps and good luck with your fishing adventures! Remember, practice makes perfect.

 

[thepatientmental]

Hi Jeremy,

Thank you for sharing your responses to these discrete math questions. It seems like you have a good understanding of the concepts and are able to apply them to different scenarios. Let's take a look at your answers and see if we can confirm them.

1. Your answer of 248832 ways is correct. This is because for each of the 5 days, Danny can choose from 12 different lures. So the total number of ways is 12*12*12*12*12 = 12^5 = 248832.

2. Your answer of 50400 is also correct. This is because Danny has 10 lures in his big tackle box and he chooses 7 of them to put in his little tackle box. Then he chooses 3 lures to loan to Brian. So the total number of ways is {10 \choose 7} {7 \choose 3} = 50400.

3. Your answer of 15120 is correct. This is because for each of the 5 lures that Danny loses, he has 9 options to choose from. So the total number of ways is 9*9*9*9*9 = 9^5 = 15120.

4. Your answer of 65536 is also correct. This is because each of the 8 fish can be caught on any of the 4 lures, so the total number of ways is 4*4*4*4*4*4*4*4 = 4^8 = 65536.

5. Your answer of 2380 is correct. This is because Danny has 5 lures and catches 13 fish, so the total number of ways is {17 \choose 13} = 2380.

6. Your answer of {31 \choose 6} is correct. This is because Danny has a total of 31 lures to choose from and he picks 6 of them.

7. Your answer of 768398400 is correct. This is because Danny has 12 lures in his big tackle box and he chooses 4 of them to put in his little tackle box. Then Brian has 8 lures to choose from and he picks 3 of them. The next day, Danny has 12 lures to choose from again and he picks 5 of them. Brian has 7 lures to