Counting problems

3. ADDITIONAL PRACTICE PROBLEMS

Seven people are going to a party: Alice, Bob, Carl, Diane, Eve, Frank, and George. When they have all arrived, everyone shakes hands.
Q1: How many handshakes were there altogether?

They then go to the table to eat, but they can't agree on a seating arrangement.
Q2: How many possibilities are there if Alice always stays at the head (since it is her birthday)?

Q3: How many possibilities are there if Alice can sit anywhere?

They decide to play bridge after dinner. When the cards have been dealt, Carl says that he thinks he got the same hand as last time.
Q4: What is the likelihood that he is right?

Finally, they decide to play chess. Alice just wants to watch, and sets up three boards.
Q5: How many different ways can they be matched?[/i]

Where do you find yourself stuck? What have you tried?

sara_bushra99 said:

Seven people are going to a party: Alice, Bob, Carl, Diane, Eve, Frank, and George. When they have all arrived, everyone shakes hands.
Q1: How many handshakes were there altogether?

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Have you been given a formula for this? If so, plug into that. If not, draw a circle, and draw seven dots (labelled "A", "B", ..., "G") for the seven people standing around shaking hands. Then start with A (that is, with Alice) and draw lines from A to each of the other dots, representing the people with whom Alice shook hands. How many lines is that? Then go to B. B has already shaken hands with A. How many lines are left to draw for B? Then for C? D? And so forth. When you're done, count up the number of lines.

sara_bushra99 said:

They then go to the table to eat, but they can't agree on a seating arrangement.
Q2: How many possibilities are there if Alice always stays at the head (since it is her birthday)?

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One position of seven is taken. How many positions are left? How many options for each position are left? Will the arrangements be combinations (where position doesn't matter) or permutations (where position does matter)? Apply the appropriate formula.

sara_bushra99 said:

Q3: How many possibilities are there if Alice can sit anywhere?

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In this case, no position is fixed. Apply the same reasoning as in Q2 to the new number of positions and options for each position.

Please reply showing your work so far on the first three questions. Thank you!