If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is... Problem : 
If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is
(a) $2^6$ (b)  $2^{12}$ (c) $2^{16}$ (d)$2^{20}$
My approach : 
$f(x) = ^{7-x}C_x   =  \frac{(7-x)!}{x!(7-2x)!} $
But having no idea how to find the range of this. 
Let R be a binary relation on a set A.
R is reflexive, iff for all x $\in$  A, (x,x) $\in$ R, i.e. xRx is true.
Examples:
Equality is a reflexive relation

for any object x,  x = x is true.

Please help in finding the range and no. of reflexive relation of this will be of great help thanks.

Assuming that $x \in \mathbb{Z}$. The binomial coefficient $\binom{7-x}{x}$ is defined for $0 \leq x \leq 7$ and $x \leq 7-x$. Thus $x \in \{0,1,2,3\}$. Now you can find possible values of this coefficient. And this gives the size of set $A$. 
Any reflexive relation $R$ should have pairs of the form $(x,x)$ for all $x \in A$. So if $|A|=n$, then $R$ should have these $n$ pairs. For "other" $n^2-n$ pairs from $A \times A$, either they are in $R$ or they are not. So each such pair's fate can be decided in $2$ ways. This will help you with the number of reflexive relations.
