If $^{2n}C_2 : ^nC_2 = 9:2$ and $^nC_r = 10$,then $r = $

$A$ question paper has two sections $A$ and $B$,in which section-$A$ has $8$ questions and section-$B$ has $6$ questions. $A$ student has to answer a total of $10$ questions,choosing at least $4$ questions from section-$A$ and at least $3$ questions from section-$B$. The number of ways a student can answer the paper is:

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements is:

From a class of $25$ students,$10$ are to be chosen for an excursion party. There are $3$ students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Six '$+$' and four '$-$' signs are to be placed in a straight line so that no two '$-$' signs come together. Then the total number of ways is:

If $^{15}C_{3r} = ^{15}C_{r+3}$,find the value of $r$.