Inversion Probability.cpp

#include <bits/stdc++.h>
using namespace std;
typedef long long LL;

struct bigint {
	vector<int> a;
	int sign;
	static const int base_digits = 9;
	static const int base = 1000000000;
	int size() {
		if (a.empty()) return 0;
		int ans = (a.size() - 1) * base_digits;
		int ca = a.back();
		while (ca)
			ans++, ca /= 10;
		return ans;
	}
	bigint operator^(const bigint &v) {
		bigint ans = 1, a = *this, b = v;
		while (!b.isZero()) {
			if (b % 2)
				ans *= a;
			a *= a, b /= 2;
		}
		return ans;
	}
	string to_string() {
		stringstream ss;
		ss << *this;
		string s;
		ss >> s;
		return s;
	}
	int sumof() {
		string s = to_string();
		int ans = 0;
		for (auto c : s) ans += c - '0';
		return ans;
	}
	bigint() : sign(1) {
	}

	bigint(long long v) {
		*this = v;
	}

	bigint(const string &s) {
		read(s);
	}

	void operator=(const bigint &v) {
		sign = v.sign;
		a = v.a;
	}

	void operator=(long long v) {
		sign = 1;
		a.clear();
		if (v < 0)
			sign = -1, v = -v;
		for (; v > 0; v = v / base)
			a.push_back(v % base);
	}

	bigint operator+(const bigint &v) const {
		if (sign == v.sign) {
			bigint res = v;

			for (int i = 0, carry = 0; i < (int)max(a.size(), v.a.size()) || carry; ++i) {
				if (i == (int)res.a.size())
					res.a.push_back(0);
				res.a[i] += carry + (i < (int)a.size() ? a[i] : 0);
				carry = res.a[i] >= base;
				if (carry)
					res.a[i] -= base;
			}
			return res;
		}
		return *this - (-v);
	}

	bigint operator-(const bigint &v) const {
		if (sign == v.sign) {
			if (abs() >= v.abs()) {
				bigint res = *this;
				for (int i = 0, carry = 0; i < (int)v.a.size() || carry; ++i) {
					res.a[i] -= carry + (i < (int)v.a.size() ? v.a[i] : 0);
					carry = res.a[i] < 0;
					if (carry)
						res.a[i] += base;
				}
				res.trim();
				return res;
			}
			return -(v - *this);
		}
		return *this + (-v);
	}

	void operator*=(int v) {
		if (v < 0)
			sign = -sign, v = -v;
		for (int i = 0, carry = 0; i < (int)a.size() || carry; ++i) {
			if (i == (int)a.size())
				a.push_back(0);
			long long cur = a[i] * (long long)v + carry;
			carry = (int)(cur / base);
			a[i] = (int)(cur % base);
		}
		trim();
	}

	bigint operator*(int v) const {
		bigint res = *this;
		res *= v;
		return res;
	}

	void operator*=(long long v) {
		if (v < 0)
			sign = -sign, v = -v;
		if (v > base) {
			*this = *this * (v / base) * base + *this * (v % base);
			return;
		}
		for (int i = 0, carry = 0; i < (int)a.size() || carry; ++i) {
			if (i == (int)a.size())
				a.push_back(0);
			long long cur = a[i] * (long long)v + carry;
			carry = (int)(cur / base);
			a[i] = (int)(cur % base);
		}
		trim();
	}

	bigint operator*(long long v) const {
		bigint res = *this;
		res *= v;
		return res;
	}

	friend pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
		int norm = base / (b1.a.back() + 1);
		bigint a = a1.abs() * norm;
		bigint b = b1.abs() * norm;
		bigint q, r;
		q.a.resize(a.a.size());

		for (int i = a.a.size() - 1; i >= 0; i--) {
			r *= base;
			r += a.a[i];
			int s1 = r.a.size() <= b.a.size() ? 0 : r.a[b.a.size()];
			int s2 = r.a.size() <= b.a.size() - 1 ? 0 : r.a[b.a.size() - 1];
			int d = ((long long)base * s1 + s2) / b.a.back();
			r -= b * d;
			while (r < 0)
				r += b, --d;
			q.a[i] = d;
		}

		q.sign = a1.sign * b1.sign;
		r.sign = a1.sign;
		q.trim();
		r.trim();
		return make_pair(q, r / norm);
	}

	bigint operator/(const bigint &v) const {
		return divmod(*this, v).first;
	}

	bigint operator%(const bigint &v) const {
		return divmod(*this, v).second;
	}

	void operator/=(int v) {
		if (v < 0)
			sign = -sign, v = -v;
		for (int i = (int)a.size() - 1, rem = 0; i >= 0; --i) {
			long long cur = a[i] + rem * (long long)base;
			a[i] = (int)(cur / v);
			rem = (int)(cur % v);
		}
		trim();
	}

	bigint operator/(int v) const {
		bigint res = *this;
		res /= v;
		return res;
	}

	int operator%(int v) const {
		if (v < 0)
			v = -v;
		int m = 0;
		for (int i = a.size() - 1; i >= 0; --i)
			m = (a[i] + m * (long long)base) % v;
		return m * sign;
	}

	void operator+=(const bigint &v) {
		*this = *this + v;
	}
	void operator-=(const bigint &v) {
		*this = *this - v;
	}
	void operator*=(const bigint &v) {
		*this = *this * v;
	}
	void operator/=(const bigint &v) {
		*this = *this / v;
	}

	bool operator<(const bigint &v) const {
		if (sign != v.sign)
			return sign < v.sign;
		if (a.size() != v.a.size())
			return a.size() * sign < v.a.size() * v.sign;
		for (int i = a.size() - 1; i >= 0; i--)
			if (a[i] != v.a[i])
				return a[i] * sign < v.a[i] * sign;
		return false;
	}

	bool operator>(const bigint &v) const {
		return v < *this;
	}
	bool operator<=(const bigint &v) const {
		return !(v < *this);
	}
	bool operator>=(const bigint &v) const {
		return !(*this < v);
	}
	bool operator==(const bigint &v) const {
		return !(*this < v) && !(v < *this);
	}
	bool operator!=(const bigint &v) const {
		return *this < v || v < *this;
	}

	void trim() {
		while (!a.empty() && !a.back())
			a.pop_back();
		if (a.empty())
			sign = 1;
	}

	bool isZero() const {
		return a.empty() || (a.size() == 1 && !a[0]);
	}

	bigint operator-() const {
		bigint res = *this;
		res.sign = -sign;
		return res;
	}

	bigint abs() const {
		bigint res = *this;
		res.sign *= res.sign;
		return res;
	}

	long long longValue() const {
		long long res = 0;
		for (int i = a.size() - 1; i >= 0; i--)
			res = res * base + a[i];
		return res * sign;
	}

	friend bigint gcd(const bigint &a, const bigint &b) {
		return b.isZero() ? a : gcd(b, a % b);
	}
	friend bigint lcm(const bigint &a, const bigint &b) {
		return a / gcd(a, b) * b;
	}

	void read(const string &s) {
		sign = 1;
		a.clear();
		int pos = 0;
		while (pos < (int)s.size() && (s[pos] == '-' || s[pos] == '+')) {
			if (s[pos] == '-')
				sign = -sign;
			++pos;
		}
		for (int i = s.size() - 1; i >= pos; i -= base_digits) {
			int x = 0;
			for (int j = max(pos, i - base_digits + 1); j <= i; j++)
				x = x * 10 + s[j] - '0';
			a.push_back(x);
		}
		trim();
	}

	friend istream &operator>>(istream &stream, bigint &v) {
		string s;
		stream >> s;
		v.read(s);
		return stream;
	}

	friend ostream &operator<<(ostream &stream, const bigint &v) {
		if (v.sign == -1)
			stream << '-';
		stream << (v.a.empty() ? 0 : v.a.back());
		for (int i = (int)v.a.size() - 2; i >= 0; --i)
			stream << setw(base_digits) << setfill('0') << v.a[i];
		return stream;
	}

	static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
		vector<long long> p(max(old_digits, new_digits) + 1);
		p[0] = 1;
		for (int i = 1; i < (int)p.size(); i++)
			p[i] = p[i - 1] * 10;
		vector<int> res;
		long long cur = 0;
		int cur_digits = 0;
		for (int i = 0; i < (int)a.size(); i++) {
			cur += a[i] * p[cur_digits];
			cur_digits += old_digits;
			while (cur_digits >= new_digits) {
				res.push_back(int(cur % p[new_digits]));
				cur /= p[new_digits];
				cur_digits -= new_digits;
			}
		}
		res.push_back((int)cur);
		while (!res.empty() && !res.back())
			res.pop_back();
		return res;
	}

	typedef vector<long long> vll;

	static vll karatsubaMultiply(const vll &a, const vll &b) {
		int n = a.size();
		vll res(n + n);
		if (n <= 32) {
			for (int i = 0; i < n; i++)
				for (int j = 0; j < n; j++)
					res[i + j] += a[i] * b[j];
			return res;
		}

		int k = n >> 1;
		vll a1(a.begin(), a.begin() + k);
		vll a2(a.begin() + k, a.end());
		vll b1(b.begin(), b.begin() + k);
		vll b2(b.begin() + k, b.end());

		vll a1b1 = karatsubaMultiply(a1, b1);
		vll a2b2 = karatsubaMultiply(a2, b2);

		for (int i = 0; i < k; i++)
			a2[i] += a1[i];
		for (int i = 0; i < k; i++)
			b2[i] += b1[i];

		vll r = karatsubaMultiply(a2, b2);
		for (int i = 0; i < (int)a1b1.size(); i++)
			r[i] -= a1b1[i];
		for (int i = 0; i < (int)a2b2.size(); i++)
			r[i] -= a2b2[i];

		for (int i = 0; i < (int)r.size(); i++)
			res[i + k] += r[i];
		for (int i = 0; i < (int)a1b1.size(); i++)
			res[i] += a1b1[i];
		for (int i = 0; i < (int)a2b2.size(); i++)
			res[i + n] += a2b2[i];
		return res;
	}

	bigint operator*(const bigint &v) const {
		vector<int> a6 = convert_base(this->a, base_digits, 6);
		vector<int> b6 = convert_base(v.a, base_digits, 6);
		vll a(a6.begin(), a6.end());
		vll b(b6.begin(), b6.end());
		while (a.size() < b.size())
			a.push_back(0);
		while (b.size() < a.size())
			b.push_back(0);
		while (a.size() & (a.size() - 1))
			a.push_back(0), b.push_back(0);
		vll c = karatsubaMultiply(a, b);
		bigint res;
		res.sign = sign * v.sign;
		for (int i = 0, carry = 0; i < (int)c.size(); i++) {
			long long cur = c[i] + carry;
			res.a.push_back((int)(cur % 1000000));
			carry = (int)(cur / 1000000);
		}
		res.a = convert_base(res.a, 6, base_digits);
		res.trim();
		return res;
	}
};

int main() {
	ios_base::sync_with_stdio(false);
	int n;
	cin >> n;
	vector<int> v(n), p, q;
	bigint d = 1, num = 0;
	for (auto &x : v) cin >> x;
	for (int i = 0; i < n; i++) {
		for (int j = i + 1; j < n; j++) {
			int x;
			if (v[i] <= v[j]) {
				x = (0 + v[i] - 1) * v[i] / 2;
			} else {
				x = (0 + v[j] - 1) * v[j] / 2;
				x += v[j] * (v[i] - v[j]);
			}
			p.push_back(x);
			q.push_back(v[j] * v[i]);
			d *= v[j] * v[i];
		}
	}
	for (unsigned i = 0; i < p.size(); i++) {
		num += (d / q[i]) * p[i];
	}
	bigint firstpart = num / d;
	vector<bigint> nr;
	num = num % d;
	for (int i = 0; i < 7; i++) {
		num *= 10;
		nr.push_back(num / d);
		num = num % d;
	}
	if ((nr[6] == 5 && num > 0) || nr[6] > 5 || (nr[6] == 5 && nr[5] % 2 == 1)) {
		nr[5] += 1;
	}
	for (int i = 6; i > 0; i--) {
		if (nr[i] == 10) {
			nr[i] = 0;
			nr[i - 1] += 1;
		}
	}
	if (nr[0] == 10) {
		firstpart += 1;
		nr[0] = 0;
	}
	cout << firstpart << ".";
	for (int i = 0; i < 6; i++) {
		cout << nr[i];
	}
	cout << "\n";
	return 0;
}