A set contains $(2n + 1)$ elements. The number of sub-sets of the set which contains at most $n$ elements is :-
$2^{n-1}$
$2^{n+1}$
$2^{2n}$
$2^n$
The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by
The number of ways in which an examiner can assign $30$ marks to $8$ questions, giving not less than $2$ marks to any question, is
The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is
Let $S=\{1,2,3, \ldots ., 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_K$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$
Determine $n$ if
$^{2 n} C_{3}:\,^{n} C_{3}=12: 1$