Help with some Permutations Problems

[rudram592]

Everyone at the party shook hands with everyone else exactly once. If there were a total of

handshakes, how many people were at the party?



Kathy tossed a coin

times and got

heads and

tails. How many different sequences of results could she have gotten?



There are

teams in a basketball league. Eight teams are in the East conference, and the other eight teams are in the West conference. If a team plays each other team from the same conference

times and from the difference conference twice, how many games are there in total?


Thanks! These are some questions that have stumped me over the last few weeks

 

[Romsek]

1) Think about how many distinct pairs you can make from N people. This must equal 21

Person N shakes everyone's hand and leaves. They shake (N-1) hands.
Person N-1 does the same shaking now (N-2) hands.
This occurs until Person 2 shakes Person 1's hand.

So \(\displaystyle 1 + 2 + ... +(N-1) = 21\)
\(\displaystyle N(N-1) = 42\\N=7\)
Another way to look at this problem is that you have N people and you need to make unique
pairs where order within the pair doesn't matter. Thus \(\displaystyle \dbinom{N}{2}=21\)
and it's not too hard to solve that \(\displaystyle N=7\) from there.



If I can place the heads, the tails just take up the remaining slots. How many ways can I place the heads?

There are \(\displaystyle \dbinom{8}{3}=56\) ways to place the 3 heads. The 5 tails fill in the remaining slots.


Just use the idea from the first question. Now the "shake hands (play basketball)" 4 times and 2 times instead of just once.

\(\displaystyle N=\dbinom{8}{2}\cdot 4 + \dbinom{8}{2}\cdot 2 = 3\cdot 56 = 168 \text{ total games}\)

 

[pka]

1) Everyone at the party shook hands with everyone else exactly once. If there were a total of

handshakes, how many people were at the party?
2) Kathy tossed a coin

times and got

heads and

tails. How many different sequences of results could she have gotten?
3) There are

teams in a basketball league. Eight teams are in the East conference, and the other eight teams are in the West conference. If a team plays each other team from the same conference

times and from the difference conference twice, how many games are there in total?

1) \(\displaystyle \dbinom{N}{2}=21\).

2) How many places out of eight can five heads occupy?

3) \(\displaystyle \dbinom{8}{2}=~?\) But what else needs doing?