math help please

Question

In triangle ABC, if ∠.ABC=60 degrees ,CT vertical line from point C,AT = square root 3,and AT =P,then the length of line BC is...

A.1/6p*(squareroot 6)
B.1/3p*(squareroot 6)
C.1/2p*(squareroot 6)
D.2/3p*(squareroot 6)
E.p*(squareroot 6)

 

 

 

A badminton team consists of 5 players.The coach selects 2 players to play in singles and 4 players in doubles.If the rule allows a single player to play in doubles once,then the number of choice which can be made is...

 

A.240

B.120

C.80

D.60

E.30

Answer 1

 

We can first calculate the number of choices for the doubles, by observing that we need only to omit one player and pair up the rest.  So we have 5 choices for the four doubles players, and for each of these 5 choices, we can pair them in 4Choose2 ways.  That is if we have A,B,C,D on our doubles squad, we can create the teams by selecting any two of them for the first team, and by default will have selected the second team...

 

the choices are:

1. AB, CD

2. AC, BD

3. AD, BC

 

 

Namely A can be partnered with any of the other 3 people on the 4 man doubles squad.  Thus for each of the 5 choices of doubles squads there are exactly 3 choices for the pairing of teammates...this gives 15 choices for the coach filling out the doubles roster.  

 

Next we need to determine who plays in the singles events...we necessarily must assign the unchosen player to singles, and pick one of the 4 members from the doubles squad to play in both events...this gives 4 choices for each one of our 15 possible ways to pick the doubles teams, leading to 15*4=60 total choices for how to assign the players to the rosters.

 

Hope that helps!