SOLUTION: A student has to choose an elective class for the last semester of high school. Her counselor suggested 9 electives, 7 of which were technology classes. If the student randomly ch

Algebra.Com
Question 1201830: A student has to choose an elective class for the last semester of high school. Her counselor suggested 9 electives, 7 of which were technology classes.
If the student randomly chose to read about 6 of the electives, what is the probability that all of them are technology classes?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52805)   (Show Source): You can put this solution on YOUR website!
.
A student has to choose an elective class for the last semester of high school.
Her counselor suggested 9 electives, 7 of which were technology classes.
If the student randomly chose to read about 6 of the electives,
what is the probability that all of them are technology classes?
~~~~~~~~~~~~~~~~ 

In all, there are   =  = 84 of all possible different 
sextuples from 9 possible classes.


Of them, the number of all different possible sextuples  
comprised of technology classes is   = 7.


The probability the problem asks for, is  P =  =  = .    ANSWER

Solved.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 1/12
1/12 = 0.083333 approximately

================================================================================

Explanation:

Let's find out how many ways there are to pick 6 technology classes to read about.
n = 7 technology classes total
r = 6 slots to fill
The order of the classes doesn't matter.
Use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
7 C 6 = (7!)/(6!*(7-6)!)
7 C 6 = (7!)/(6!*1!)
7 C 6 = (7*6!)/(6!*1!)
7 C 6 = (7)/(1!)
7 C 6 = 7/1
7 C 6 = 7
There are 7 ways to pick 6 technology classes to read about.

That's a fairly long-winded approach.
Is there a shortcut, and a way to avoid that formula?
Yes there is.
Imagine there are 2 groups: A and B
Group A represents what you pick (those 6 items to read).
Group B is what you don't pick: that 1 class left out.
At this point it should be fairly clear that there are 7 ways to fill group B, which means there are 7 ways to fill group A.

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

Now let's determine how many ways there are to select 6 classes, where some of which may not be technology classes.
n = 9 classes total
r = 6 slots
Use the nCr formula
n C r = (n!)/(r!(n-r)!)
9 C 6 = (9!)/(6!*(9-6)!)
9 C 6 = (9!)/(6!*3!)
9 C 6 = (9*8*7*6!)/(6!*3!)
9 C 6 = (9*8*7)/(3!)
9 C 6 = (9*8*7)/(3*2*1)
9 C 6 = 504/6
9 C 6 = 84
There are 84 ways to select 6 classes from a candidate pool of 9.

Rephrased another way:
There are 9*8*7 = 504 permutations
We divide by 3*2*1 = 6 to adjust the fact order doesn't matter (something like ABC is the same as BAC).
There are 6 ways to arrange any trio of items.
Therefore, we arrive at 504/6 = 84 combinations.

Each nCr value can be found in Pascal's Triangle.
For instance, the value 84 is in the row that starts with 1,9,...

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

There's a lot to take in, so read each subsection again to make sure you understand what is going on.

To recap:Order does not matter.

Divide those values to get the probability we want.
7/84 = (1*7)/(12*7) = 1/12 is the probability that all 6 selections are technology classes.

Use a calculator to get its approximate value
1/12 = 0.083333
It might be better to stick to the fraction form since it is exact.
Be sure to follow your teacher's instructions.
RELATED QUESTIONS

A student has to choose an elective class for the last semester of high school. His... (answered by Edwin McCravy)
A high school student needs 9 more classes to graduate. If she has met the prerequisites (answered by Alan3354)
Next school year, 4 out of every 9 students plan to take an elective computer course. If... (answered by rfer)
Every year, students have to decide which four classes to take each semester. They must... (answered by Boreal)
An English class can choose from 6 books to read this semester. How many different ways... (answered by checkley77)
Next school year, 2 out of every 7 students plan to take an elective computer course. If (answered by mananth)
Of the students in a class, are taking the class because it is a major requirement, and (answered by greenestamps)
A student is filling in her schedule. She has 3 spots left open for classes on her... (answered by stanbon)
A high school math teacher wants to assess the effectiveness of a new teaching method... (answered by Theo)