Out of $6$ books, in how many ways can a set of one or more books be chosen
$64$
$63$
$62$
$65$
(b) Required number of ways
$=^6{C_1}+^6{C_2}+^6{C_3}+^6{C_4}+^6{C_5}+^6{C_6}=2^6-1=63$.
A student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select one book is $63$, then the value of $n$ is
A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no comman friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is :
If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$, then the value of $r$ is equal to:
A set contains $(2n + 1)$ elements. The number of sub-sets of the set which contains at most $n$ elements is :-
The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples, is
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