Contest Duration: - (local time) (110 minutes) Back to Home
E - BBQ Hard /

Time Limit: 2 sec / Memory Limit: 256 MB

### 問題文

セットを 2 個選び、セット 2 つに含まれる全ての具材を好きな順番で串 2 本に刺すことを考えます。 このとき、作ることの出来る串焼きは何通り考えられるでしょうか？ ただし、串どうしは区別でき、肉どうしや野菜どうしは区別できないものとします。 答えは非常に大きな数になる可能性があるので、10^9+7 で割った余りを求めてください。

### 制約

• 2≦N≦200,000
• 1≦A_i≦2000, 1≦B_i≦2000

### 入力

N
A_1 B_1
A_2 B_2
:
A_N B_N


### 入力例 1

3
1 1
1 1
2 1


### 出力例 1

26


Score : 1400 points

### Problem Statement

Snuke is having another barbeque party.

This time, he will make one serving of Skewer Meal.

He has a stock of N Skewer Meal Packs. The i-th Skewer Meal Pack contains one skewer, A_i pieces of beef and B_i pieces of green pepper. All skewers in these packs are different and distinguishable, while all pieces of beef and all pieces of green pepper are, respectively, indistinguishable.

To make a Skewer Meal, he chooses two of his Skewer Meal Packs, and takes out all of the contents from the chosen packs, that is, two skewers and some pieces of beef or green pepper. (Remaining Skewer Meal Packs will not be used.) Then, all those pieces of food are threaded onto both skewers, one by one, in any order.

(See the image in the Sample section for better understanding.)

In how many different ways can he make a Skewer Meal? Two ways of making a Skewer Meal is different if and only if the sets of the used skewers are different, or the orders of the pieces of food are different. Since this number can be extremely large, find it modulo 10^9+7.

### Constraints

• 2≦N≦200,000
• 1≦A_i≦2000, 1≦B_i≦2000

### Input

The input is given from Standard Input in the following format:

N
A_1 B_1
A_2 B_2
:
A_N B_N


### Output

Print the number of the different ways Snuke can make a serving of Skewer Meal, modulo 10^9+7.

### Sample Input 1

3
1 1
1 1
2 1


### Sample Output 1

26


The 26 ways of making a Skewer Meal are shown below. Gray bars represent skewers, each with a number denoting the Skewer Meal Set that contained the skewer. Brown and green rectangles represent pieces of beef and green pepper, respectively.