FIRST WE MUST FIND THE TOTAL NUMBER OF 5 MEMBER TEAMS, WITH OUR WI...

MANHATTAN TEST 2 ANSWERS

30. First we must find the total number of 5 member teams, with

our without John and Peter. We can solve this using an anagram

model in which each of the 9 players (A – I) is assigned either a Y

(for being chosen) or an N (for not being chosen):

Player A B C D

E F

H I

Chosen ? Y

Y Y Y

Y N N

N N

It is the various arrangements of Y’s and N’s above that would yield

all of the different combinations, so we can find the number of

possible teams here by considering how many anagrams of

YYYYYNNNN exist:

5! 4!

9 × 8 × 7 × 6 × 5

5 × 4 × 3 × 2 × 1

= (3 × 7 × 6) = 126

(because there are 9! ways to order 9 objects)

(because the 5Y's and 4N's are identical)

So there are 126 possible teams of 5. Since the question asks for

the probability of choosing a team that includes John and Peter, we

need to determine how many of the 126 include John and Peter. If

we reserve two of the 5 spots on a team for John and Peter, there

will be 3 spots left, which must be filled by 3 of the remaining 7

players (remember John and Peter were already selected).

Therefore the number of teams including John and Peter will be

equal to the number of 3-player teams that can be formed from a 7-

player pool. We can approach the problem as we did above:

Player A

C D

Chosen Y

Y N

The number of possible YYYNNNN anagrams is:

3 × 2 ×1

= 35

3! 4!

7 × 6 × 5

Since 35 of the total possible 126 teams include John and Peter, the

probability of selecting a team with both John and Peter is 35/126 or