Count the Number of Infection Sequences
<p>You are given an integer <code>n</code> and an array <code>sick</code> sorted in increasing order, representing positions of infected people in a line of <code>n</code> people.</p>

<p>At each step, <strong>one </strong>uninfected person <strong>adjacent</strong> to an infected person gets infected. This process continues until everyone is infected.</p>

<p>An <strong>infection sequence</strong> is the order in which uninfected people become infected, excluding those initially infected.</p>

<p>Return the number of different infection sequences possible, modulo <code>10<sup>9</sup>+7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 5, sick = [0,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

<p><strong>Explanation:</strong></p>

<p>There is a total of 6 different sequences overall.</p>

<ul>
	<li>Valid infection sequences are <code>[1,2,3]</code>, <code>[1,3,2]</code>, <code>[3,2,1]</code> and <code>[3,1,2]</code>.</li>
	<li><code>[2,3,1]</code> and <code>[2,1,3]</code> are not valid infection sequences because the person at index 2 cannot be infected at the first step.</li>
</ul>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 4, sick = [1]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>There is a total of 6 different sequences overall.</p>

<ul>
	<li>Valid infection sequences are <code>[0,2,3]</code>, <code>[2,0,3]</code> and <code>[2,3,0]</code>.</li>
	<li><code>[3,2,0]</code>, <code>[3,0,2]</code>, and <code>[0,3,2]</code> are not valid infection sequences because the infection starts at the person at index 1, then the order of infection is 2, then 3, and hence 3 cannot be infected earlier than 2.</li>
</ul>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= sick.length &lt;= n - 1</code></li>
	<li><code>0 &lt;= sick[i] &lt;= n - 1</code></li>
	<li><code>sick</code> is sorted in increasing order.</li>
</ul>
Array; Math; Combinatorics
const MX = 1e5
const MOD = 1e9 + 7

var fac [MX + 1]int

func init() {
	fac[0] = 1
	for i := 1; i <= MX; i++ {
		fac[i] = fac[i-1] * i % MOD
	}
}

func qpow(a, n int) int {
	ans := 1
	for n > 0 {
		if n&1 == 1 {
			ans = (ans * a) % MOD
		}
		a = (a * a) % MOD
		n >>= 1
	}
	return ans
}

func numberOfSequence(n int, sick []int) int {
	m := len(sick)
	nums := make([]int, m+1)

	nums[0] = sick[0]
	nums[m] = n - sick[m-1] - 1
	for i := 1; i < m; i++ {
		nums[i] = sick[i] - sick[i-1] - 1
	}

	s := 0
	for _, x := range nums {
		s += x
	}
	ans := fac[s]
	for _, x := range nums {
		if x > 0 {
			ans = ans * qpow(fac[x], MOD-2) % MOD
		}
	}
	for i := 1; i < len(nums)-1; i++ {
		if nums[i] > 1 {
			ans = ans * qpow(2, nums[i]-1) % MOD
		}
	}
	return ans
}