Let S = {1, 2, 3,....., m} , m > 3. Let $${X_1},\,....,\,{X_n}$$ be subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f (i) is the number of sets $${X_j}$$ that contain the element i. That is $$f(i) = \left\{ {j\left| i \right.\,\, \in \,{X_j}} \right\}\left| . \right.$$

Then $$\sum\limits_{i - 1}^m {f\,(i)} $$ is

For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an identity for $$f$$ if $$f\left( {a,z} \right) = a = f\left( {z,a} \right)$$ for all $$a \in N$$. Which of the following binary operations have an identify?
$${\rm I}$$) $$\,\,\,\,\,\,f\left( {x,y} \right) = x + y - 3$$
$${\rm I}{\rm I}$$ $$\,\,\,\,\,\,f\left( {x,y} \right) = {\mkern 1mu} \max \left( {x,y} \right)$$
$${\rm I}{\rm I}{\rm I}$$$$\,\,\,\,\,f\left( {x,y} \right) = \,{x^y}$$

The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Given below are four plausible reasons.

Which one of them is false?

Let $$X,. Y, Z$$ be sets of sizes $$x, y$$ and $$z$$ respectively. Let $$W = X x Y$$ and $$E$$ be the set of all subjects of $$W$$. The number of functions from $$Z$$ to $$E$$ is