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1. Everyone at the party shook hands with everyone else exactly once. If there were a total of  21 handshakes, how many people were at the party?

2. Kathy tossed a coin 8 times and got 3 heads and 5 tails. How many different sequences of results could she have gotten?

3. There are 16 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 4 times and from the difference conference twice, how many games are there in total?

 Apr 13, 2019
 #1
avatar+104650 
+1

1. Everyone at the party shook hands with everyone else exactly once. If there were a total of  21 handshakes, how many people were at the party?

 

There were 7 people

 

The 7th person shakes 6 hands

The 6th ,5 hands

The 5th, 4 hands

.

.

.

The 2nd,1 hand

The last...0 hands  (everyone else has shaken this person's hand once)

 

So

 

6 + 5 + 4 + 3 + 2 + 1  + 0  =     21

 

 

cool cool cool

 Apr 13, 2019
 #2
avatar+104650 
+1

2. Kathy tossed a coin 8 times and got 3 heads and 5 tails. How many different sequences of results could she have gotten?

 

This is the same as finding the number of identifiable "words" that can be formed from the "word"  HHHTTTTT

 

We have

 

   8!

____  =    56  identifiable "words"  or sequences

 3! 5!

 

 

 

cool cool cool

 Apr 13, 2019
 #3
avatar+104650 
+1

3. There are 16 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 4 times and from the difference conference twice, how many games are there in total?

 

 

Never sure about these....but....here's my best attempt

 

Suppose that each team plays every other team in its conference once  and every other team in the other conference once

 

Within one conference......it will be similar to the "handshake" problem

 

There will be  (8*7)/2  = 28 games withiin each conference  if each team plays each other once

So....the total number of intra-coference games will be four times this = 4 (28 + 28) = 4(56)  = 224 games

 

If each team plays each team from the other conference once there will be  (8)(8)  = 64 games

But....we must double this  = 128 games

 

So....the total number of games  =  224 + 128  =  352 games

 

 

cool cool cool

 Apr 13, 2019

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