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

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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(52794) (Show Source):
You can put this solution on YOUR website!
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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?
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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

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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.

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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,...

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

To recap:
7 ways to pick the 6 technology classes
84 ways to pick any 6 classes (technology or not)
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.