Alice invites 6 guests to her birthday party: Bob, Carol, Diane, Eve, Frank, and

Question

Alice invites 6 guests to her birthday party: Bob, Carol, Diane, Eve, Frank, and George.

A. When they arrive, they shake hands with each other. Bob asks: How many total handshakes were made?

B. When they go to the table, they argue over who sits next to whom. The only thing they do agree on is that Alice should be seated at the head of the table. So, Alice agrees to remain at the head of the table but suggests that the seating be changed every 30 minutes and that no one goes home until every possible seating has been sat. How long does this party last?

C. Suppose they take out a little time for dancing. How many guy-gal dance couples are there?

D. Frank wants to make some money and so asks his friends to pool their resources to win the lottery. In the lottery he suggests, there are 90 numbers of which a player must choose 5. How many ways can a lottery ticket be filled out? If a ticket costs $1 and the prize is $1,000,000, what advice would you give about buying lottery tickets and why?

Explanation / Answer

A. If Alice also shakes hands, the number of handshakes is 7*6/2 = 21

since each of the 7 shake hands with other 6 and each handshake is counted twice.

However if Alice does not shake hands, then the number of handshakes is 6*5/2 = 15

B. Since Alice's position is fixed, we need to find different ways in which the others sit.

This is 6! = 720 since there are 6 others.

C. There are four girls and three guys.

Each guy can pair with four girls. So there are a total of 3*4 = 12 pairs.

D. Since there are 90 numbers and every player gets to choose 5, the lottery ticket can be filled in 90C5 ways. Further since there are 7 players this can be filled out in 90C5 * 7 ways.

The probability that the first number of the lottery matches with the winning number = 1/90

So the probability that all numbers match = 1/(90)5

So it takes 905 tickets to be sure to win the prize or 905 dollars.

Since 1,000,000 = 106 < 905, it would be better not to buy any tickets.

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