Rescue Nibel!
Contest: Codeforces Round 672 (Div. 2)
Difficulty rating: 1800
Tags: combinatorics, data structures, sortings

Ori and Sein have overcome many difficult challenges. They finally lit the Shrouded Lantern and found Gumon Seal, the key to the Forlorn Ruins. When they tried to open the door to the ruins... nothing happened.

Ori was very surprised, but Sein gave the explanation quickly: clever Gumon decided to make an additional defence for the door.

There are $$$n$$$ lamps with Spirit Tree's light. Sein knows the time of turning on and off for the $$$i$$$-th lamp — $$$l_i$$$ and $$$r_i$$$ respectively. To open the door you have to choose $$$k$$$ lamps in such a way that there will be a moment of time when they all will be turned on.

While Sein decides which of the $$$k$$$ lamps to pick, Ori is interested: how many ways there are to pick such $$$k$$$ lamps that the door will open? It may happen that Sein may be wrong and there are no such $$$k$$$ lamps. The answer might be large, so print it modulo $$$998\,244\,353$$$.

Input format:
First line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 3 \cdot 10^5$$$, $$$1 \le k \le n$$$) — total number of lamps and the number of lamps that must be turned on simultaneously.

Next $$$n$$$ lines contain two integers $$$l_i$$$ ans $$$r_i$$$ ($$$1 \le l_i \le r_i \le 10^9$$$) — period of time when $$$i$$$-th lamp is turned on.

Output format:
Print one integer — the answer to the task modulo $$$998\,244\,353$$$.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In first test case there are nine sets of $$$k$$$ lamps: $$$(1, 2, 3)$$$, $$$(1, 2, 4)$$$, $$$(1, 2, 5)$$$, $$$(1, 2, 6)$$$, $$$(1, 3, 6)$$$, $$$(1, 4, 6)$$$, $$$(2, 3, 6)$$$, $$$(2, 4, 6)$$$, $$$(2, 6, 7)$$$.

In second test case $$$k=1$$$, so the answer is 3.

In third test case there are no such pairs of lamps.

In forth test case all lamps are turned on in a time $$$3$$$, so the answer is 1.

In fifth test case there are seven sets of $$$k$$$ lamps: $$$(1, 2)$$$, $$$(1, 3)$$$, $$$(2, 3)$$$, $$$(2, 4)$$$, $$$(3, 4)$$$, $$$(3, 5)$$$, $$$(4, 5)$$$.

Example input:
7 3
1 7
3 8
4 5
6 7
1 3
5 10
8 9

Example output:
9

Example input:
3 1
1 1
2 2
3 3

Example output:
3

Editorial:
In this task, we need to find the number of sets of $$$k$$$ segments such that these $$$k$$$ segments intersect at least in one point.
Let's look at the starting point of the intersection. This point will always be the beginning of a segment. Let us find the number of sets of segments that their intersection begins at the point $$$x$$$. Let us denote $$$p(x)$$$ as number of segments that pass through this point, and $$$s(x)$$$ as numbers of segments that start at this point. Then all the $$$k$$$ segments must pass through $$$x$$$ and at least one segment must start at $$$x$$$. The number of sets of segments passing through $$$x$$$ is $$${p(x) \choose k}$$$ and the number of sets of segments passing through $$$x$$$, none of which starts at $$$x$$$, is $$${p(x) - s(x) \choose k}$$$. From here we obtain that the required number of piece sets is $$${p(x) \choose k}. - {p(x) - s(x) \choose k}$$$. By summing up all possible $$$x$$$ values, we get the answer to the task.
It should be noted that $$$p(x)$$$ and $$$s(x)$$$ can be easily supported using the event method. Then, the total runtime will be $$$O(n\log n)$$$.