Question 1161853
You have a wrong formula, and formulas are not good for brain health.
For a student, understanding and reasoning is best for brain health.
You cannot raise wise students feeding them formulas.
(You can figure out by yourself how to feed babies for best brain health).
 
However, to communicate with computers, calculators, and math teachers
we need a little vocabulary/symbols. (More of that later).
The symbol {{{4!}}} represents {{{4*3*2*1=24}}} ,
a product with the first 4 counting numbers as factors.
We call it "factorial of 4" or "4 factorial".
My calculator has "x!" as a function,
but factorial is used for positive integers,
although we extend it beyond {{{1!=1}}} to {{{0!=1}}} .
 
(a) I understand that when we pick {{{4}}} students for a committee out of a class of {{{15}}} students,
we just need a group/set of {{{4}}} students.
We can list them in any order and it is the same committee
with all students equally important for now.
They can distribute their roles and responsibilities among themselves later.
We could instead make an ordered list,
where the first named is the president of the committee,
the second one is the secretary/note-taker,
the third is the treasurer,
and the last one brings the donuts to the meetings.
In that case the same 4 people mentioned in different order make a different list.
When we make lists of {{{4}}} people out of {{{15}}} available to choose from,
we have {{{15}}} choices for the first name in the list.
Afterwards, for each of those {{{15}}} choices,
we will have {{{14}}} remaining choices for a second name (the {{{14}}} not yet chosen students).
Then, we will have {{{13}}} choices for the third name,
and {{{12}}} choices for the fourth.
All in all, we could make {{{15*14*13*12=32760}}} lists.
That is not {{{32760}}} different sets of students,
because each set of 4 students can be listed {{{4*3*2*1=24}}} different ways Different orders).
We will find each different set of 4 students repeated as {{{24}}} times in those {{{32760}}} lists.
Those {{{32760}}} lists represent only
{{{(15*14*13*12)/(4*3*2*1)=32760/24=highlight(1365)}}} different committees.
 
(b) For a list of 10 students out of a class of {{{15}}} there are
{{{15*14*13*"...."*8*7*6}}} different possibilities
That is a product of {{{10}}} factors that are the numbers from 1 to 15, except the first {{{15-10=5}}} numbers.
We would tell a computer or a calculator to calculate that as computer could see that as
{{{(15*14*13*"...."*8*7*6*5*4*3*2*10)/(5*4*3*2*1)=15!/(15-10)!=15!/5!}}} .
Each set of {{{1]}}} students can be written in {{{10!}}} different orders as {{{10!}}} different lists,
so the {{{15!/(15-10)!=15!/5!}}} different lists represent
{{{15!/(15-10)!/10!=15!/5!/10!=highlight(273)}}} .
 
MORE symbols and vocabulary:
My (cheap) scientific calculator has keys for the functions {{{nPr}}} and {{{nCr}}} .
 
{{{nPr}}} (with {{{n>=r}}} , of course) represents
the number of different permutations (lists) of {{{r}}} items that can be made
out of a total of {{{n}}} different items.
It is called permutations of n taken r at a time because the same r items in different order make a different list/permutation.
The formula to calculate that number is
{{{nPr=n!/(n-r)!}}} , where
{{{n!=n*(n-1)*(n-2)*"..."*3*2*1}}} is the product of the {{{n}}} consecutive integers from {{{1}}} to {{{n}}} , and
{{{(n-r)!=n*(n-r-1)*(n-r-2)*"..."*3*2*1}}} is the product of the {{{n-r}}} consecutive integers from {{{1}}} to {{{n-r}}} ,
(except that we define {{{0!=1}}} for consistency).
{{{nPn=n!/(n-n)!=n!/0!=n!/1=n!}}} represents the number of list permutations of n items
that we can make using all n items at a time (for each list).
We just call it permutations of n, but {{{n!}}} is shorter and more direct.
 
{{{nCr=nPr/n!}}}{{{"="}}}{{{"n!/(n-r)!"/r!}}}{{{"="}}}{{{n!/(n-r)!/r!}}} represents the number of different sets/combinations you can make
by picking  {{{r}}} items out of {{{n}}} available items.
We use "C" in that symbol for combinations,
meaning that we consider {lettuce,tomato,pickle} the same
set/combination of 3 burger toppings as {tomato,pickle,lettuce} .