Question 1043933
I have been struggling to understand what the wording of the question means.
There are {{{n}} scholarships available,
and there are {{{2n+1}}} students applying for those scholarships.
Are those {{{n}}} scholarships distinguishable from each other?
Does "at least one student gets scholarship" mean {{{n>=1}}} ?
Does the wording mean that maybe not all the {{{n}}} scholarships will be awarded, but at least {{{1}}} will be awarded?
The meaning of the wording has to allow the existence of {{{63}}} different possible outcomes. I could think of only one way to make that wording work.
 
Let's say the scholarships are indistinguishable from each other, and at least {{{1}}} , but at most {{{n}}} will be awarded:
If just {{{1}}} is awarded, that is combinations of {{{2n+1}}} taken {{{1}}} at a time,
and that is obviously {{{2n+1}}} ways,
because the single scholarship could be awarded to any of the {{{2n+1}}} students.
How many different ways could it happen that exactly {{{2}}} scholarship is awarded?
That is combinations of {{{2n+1}}} taken {{{2}}} at a time,
and that is {{{(matrix(2,1,2n+1,n))=(2n+1)2n/2=(2n+1)n}}} .
Similarly we would get to a number for the case that exactly {{{3}}} , {{{4}}} , {{{"..."}}} , {{{n}}} scholarships could be awarded.
There are {{{(matrix(2,1,2n+1,n))}}}={{{(2n+1)!/((n+1)!n!)}}} possible different sets of {{{n}}} students that could be chosen to receive scholarships out of a pool of {{{2n+1}}} applicants.
Finding {{{n}}} so that the total of possible different sets of
{{{1}}} , {{{2}}} , {{{3}}} , {{{"..."}}} , or {{{n}}} scholarship recipients adds up to {{{63}}} appears complicated,
but the fact that {{{63}}} is a relatively small number makes it easy.
For {{{n=4}}} , {{{2n+1=9}}} and there are
{{{(matrix(2,1,9,4))=9*8*7*6/(2*3*4)=126}}} possible different sets of {{{4}}} students that can be chosen from {{{9}}} applicants.
Since {{{126>63}}}, we need to choose a smaller {{{n}}} .
 
It so happens that {{{highlight(n=3)}}} works.
For {{{n=3}}} , {{{2n+1=7}}} and from a pool of {{{7}}} students, there can be
{{{(matrix(2,1,7,3))=7*6*5/(2*3)=35}}} possible different sets of {{{3}}} students:
{{{(matrix(2,1,7,2))=7*6/2=21}}} possible different sets of {{{3}}} students, and
{{{7}}} sets of {{{1}}} student
that can be chosen to receive scholarships.
All in all there are {{{35+21+7=63}}} possible sets of {{{1}}} , {{{2}}}, or {{{3}}} students that can be chosen to receive scholarships out of a total of {{{7}}} student applicants.