A partial order P is defined on the set of natural numbers as following. Herw x/y denotes integer division.
i) (0, 0) $$ \in \,P$$.
ii) (a, b) $$ \in \,P$$ if and only a %
$$10\, \le $$ b % 10 and
)a/10, b/10) $$ \in \,P$$.

Consider the following ordered pairs:
$$\matrix{ {i)\,\,\,(101,\,22)} & {ii)\,\,\,(22,\,\,101)} \cr {iii)\,\,\,(145,\,\,265)} & {iv)\,\,\,(0,\,153)} \cr } $$
Which of these ordered pairs of natural numbers are comtained in P?

Consider the set S = {a, b, c, d}. Consider the following 4 partitions $$\,{\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}$$ on $$S:\,{\pi _1} = \left\{ {\overline {a\,b\,c\,d} } \right\},\,{\pi _2} = \left\{ {\overline {a\,b\,} ,\,\overline {c\,d} } \right\},\,{\pi _3} = \left\{ {\overline {a\,b\,c\,} ,\,\overline d } \right\},\,{\pi _4} = \left\{ {\overline {a\,} ,\,\overline b ,\,\overline c ,\,\overline d } \right\}.$$ Let $$ \prec $$ be the partial order on the set of partitions $$S' = \{ {\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}\} $$ defined as follows: $${\pi _i} \prec \,\,{\pi _j}$$ if and only if $${\pi _i} $$ refines $${\pi _j}$$. The poset diagram for $$(S',\, \prec )$$ is

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

Given a set of elements N = {1, 2, ....., n} and two arbitrary subsets $$A\, \subseteq \,N\,$$ and $$B\, \subseteq \,N\,$$, how many of the n! permutations $$\pi $$ from N to N satisfy $$\min \,\left( {\pi \,\left( A \right)} \right) = \min \,\left( {\pi \,\left( B \right)} \right)$$, where min (S) is the smallest integer in the set of integers S, and $${\pi \,\left( S \right)}$$ is the set of integers obtained by applying permutation $${\pi}$$ to each element of S?