A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the question?

Answer 1

#color(blue)(10)#

We can solve this using the number of combinations of #bbn# things taken #bbr# at a time.

It is necessary to decide whether you are searching for combinations or permutations when solving these kinds of problems.

Both the elements in a set and their appearance order are considered in a permutation.

Example:

The number of permutations of #ABC# taken 3 at a time is 6.

These would be:

ABC

ACB

BAC

BCA

CAB

CBA

You'll see that while the elements are the same, the order in which they appear varies amongst them.

There is only one ABC combination that can be taken three at a time, and that is:

ABC

This is so because combinations don't care about the elements' exact arrangement; instead, they focus only on the elements themselves.

We need combinations for our problem because the questions themselves are what matter—we don't care what order they are in.

The following represents combinations:

#color(white)(0)^nC_(r)#
Where #n# is the number of elements and #r# is how many at a time they are taken:
#color(white)(0)^nC_(r)=(n!)/(r!(n-r)!#
#n!# means #n(n-1)(n-2)...(n-n+2)(n-n+1)#

To clarify:

#4! =4xx3xx2xx1#
We have 5 things, so #n=5# and we are choosing 3 at a time, so #r=3#

Including these:

#(n!)/(r!(n-r)!#
#(5!)/(3!(5-3)!) =(5xx4xx3xx2xx1)/(3xx2xx1xx2xx1)#
#=(5xx4xxcancel(3)xxcancel(2)xxcancel(1))/(cancel(3)xxcancel(2)xx1xx2xxcancel(1))=(5xx4)/(1xx2)=20/2=10#
So there are #color(blue)(10)# ways of choosing the 3 questions.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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