SOLUTION: Five students will be randomly selected from the seven top students in mathematics to represent their school in a regional competition. Angela (A) and Brian (B) are among the seven

Question 1042072: Five students will be randomly selected from the seven top students in mathematics to represent their school in a regional competition. Angela (A) and Brian (B) are among the seven students, and they wish to determine the probability that they will both be chosen. Using A, B, C, D, E, F, and G to denote the students, set up a sample space to determine this probability. What is the probability they will both be chosen? Enter probability as a fraction.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Use the combination formula to get

n C r = (n!)/(r!(n-r)!)

7 C 5 = (7!)/(5!*(7-5)!)

7 C 5 = (7!)/(5!*2!)

7 C 5 = (7*6*5!)/(5!*2!)

7 C 5 = (7*6)/(2!)

7 C 5 = (7*6)/(2*1)

7 C 5 = (42)/(2)

7 C 5 = 21

So there are 21 ways to pick 5 students. The sample space would be...

A,B,C,D,E
A,B,C,D,F
A,B,C,D,G
A,B,C,E,F
A,B,C,E,G
A,B,C,F,G
A,B,D,E,F
A,B,D,E,G
A,B,D,F,G
A,B,E,F,G
A,C,D,E,F
A,C,D,E,G
A,C,D,F,G
A,C,E,F,G
A,D,E,F,G
B,C,D,E,F
B,C,D,E,G
B,C,D,F,G
B,C,E,F,G
B,D,E,F,G
C,D,E,F,G

If you counted all the possibilities in the sample space above, you'd count out 21 different items. Each item has its own line.

--------------------------------------------------------------------------------------

Now let's lock up the first two slots for students A and B. That leaves 7-2 = 5 students left and 5-2 = 3 slots left

So using the combination formula again gives

n C r = (n!)/(r!(n-r)!)

5 C 3 = (5!)/(3!*(5-3)!)

5 C 3 = (5!)/(3!*2!)

5 C 3 = (5*4*3!)/(3!*2!)

5 C 3 = (5*4)/(2!)

5 C 3 = (5*4)/(2*1)

5 C 3 = (20)/(2)

5 C 3 = 10

There are 10 ways to pick a group of five students (from the pool of seven) where students A and B are guaranteed to go.

The 10 different ways are listed below

A,B,C,D,E
A,B,C,D,F
A,B,C,D,G
A,B,C,E,F
A,B,C,E,G
A,B,C,F,G
A,B,D,E,F
A,B,D,E,G
A,B,D,F,G
A,B,E,F,G

That list of 10 is a subset of the first list of items shown above.

--------------------------------------------------------------------------------------

Now divide the two values to get 10/21 which is the final answer

Side note: 10/21 = 0.47619047619048 = 47.619047619048%

--------------------------------------------------------------------------------------

Final Answer: 10/21

RELATED QUESTIONS
Five students will be randomly selected from the seven top students in mathematics to... (answered by Fombitz)

Consider the following counting problem. At a local private high school there are 26... (answered by lostin)

In a survey of randomly selected students, students chose their preference from among... (answered by ikleyn)

How many different sets of 4 students can be chosen out of 17 qualified students to... (answered by Alan3354)

In how many ways can nine students be selected randomly from a class of 40... (answered by solver91311)

In how many 5-digit numbers can be formed from the 10 digits (0-9), if a) Repetitions... (answered by Edwin McCravy)

i) In how many 5-digit numbers can be formed from the 10 digits (0-9), if a) Repetitions (answered by ikleyn)

My question is.... A survey taken during class revealed 30% of the students in a class... (answered by stanbon)

The names of four students are placed in a small box and two will be selected to... (answered by Edwin McCravy,Theo,ikleyn)