[Solved] MA2631 Conference 1

1. In how many ways can 3 science fiction books, 4 math books and 1 cooking book arranged on abookshelf, if
   • there are no restrictions on the arrangement?
   • all the science fiction books have to be stored together and also all the math books have to bestored together?
   • (only) the math books have to be stored together, the rest can be arranged without restriction?
2. Mike has nine friends and wants to throw a party. As his appartment is not big enough to invite all ofthem, he decides to invite only six.
   • How many choices of invitations has he?
   • Two of his friends of his are feuding and will not attend the party together. Accounting for thisfact, how many possibilities has he?
   • Two of his friends are very close and will attend the party only if invited together. Accountingfor this fact, how many possibilities has he?
3. A coin is tossed repeatedly until the first time heads appears.
   • Describe mathematically the sample space of this experiment.
   • Describe mathematically the events
     • = there are no more than four tails
     • = there are at least two tails
   • Describe mathematically the events E F and E F^c
4. Given a family of events E₁,E₂,,E_n, on some sample space , construct a new family F₁,F₂,,F_n, on the same sample space such that the events F_i are monotone, (F_m F_n for m n)and

n n [ [

F_k = for any positive integer n.

k=1 k=1

Prove that the constructed family has the desired properties.

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