Discrete Maths

Problem 1

("Warmup. There is a saying: "If you really know it, you can explain it to your grandparents". Pretend that you're having coffee with your grandparents. Explain the the concepts permutations and combinations, and the difference between the two. Give examples to support your argument.")

Problem 2

1. ("In how many different ways can the participants win the prizes? Show a formula and compute a final answer showing intermediate computations.")
2. ("Let us assume 1 that it has been decided beforehand that a particular participant is to get a prize. In how many different ways can the participants now win the prizes? Show a formula and compute a final answer showing intermediate computations.")

Problem 3

("A firm employees 25 computer scientists, 20 engineers and 10 technicians.")

1. ("In how many different ways can we select 3 employees for a committee, if they must all come from the same profession (for example all computer scientists)? Find a formula and compute a final answer showing intermediate results.")
2. ("Let us now select 8 employees at random to serve on a committee. Find the probability that it contains employees of exactly two of these professions. (For example one or more computer scientists and one or more engineers but no technician.) Show a formula but you do not have to compute a final answer.")

Problem 4

("A runner has 3 colours of nail polish to colour his/her fingernails. Each nail is coloured in one colour.")

1. ("In how many different ways can this be done? Show a formula and compute a final answer.")
2. ("In how many different ways can the nails be painted, if every nail on her right hand must have the same colour? Show a formula and compute a final answer.")
3. ("If the runner picks colours at random, find the probability that there are at least 2 colours on each hand. Show a formula and compute a final answer.")

Problem 5

("Jn and Gunna are dealt 5 cards each from a standard deck of 52 cards. ")

1. ("In how many ways can it happen that Jn obtains 3 clubs and 2 spades and Gunna obtains 4 diamonds and one heart? Show a formula and calculate a final answer showing intermediate results.")
2. ("Find the probability that Jn obtains 5 cards of one suit and Gunna also obtains 5 cards of one suit. (The suits are hearts, spades, diamonds and clubs.) Show a formula but you do not have to calculate a final answer.")

Problem 6

("Let us consider codewords, that are a string of letters from the set A,B,C and digits from the set 1,2,3,4,5. A codeword is valid if it contains an odd number of letters (f.ex. 5B4AB andBCBCC are valid codewords, but 233 and 13BA are invalid). Let an be the number of valid codewords of length n.")

1. "Find a₁and a₂.
2. Find a recurrence relation for a_n. Argue carefully.
3. "Use the recurrence relation to find a₄. Show calculations.