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
mod = 10**9 + 7
mx = 10**5
fac = [1] * (mx + 1)
for i in range(2, mx + 1):
    fac[i] = fac[i - 1] * i % mod


class Solution:
    def numberOfSequence(self, n: int, sick: List[int]) -> int:
        nums = [b - a - 1 for a, b in pairwise([-1] + sick + [n])]
        ans = 1
        s = sum(nums)
        ans = fac[s]
        for x in nums:
            if x:
                ans = ans * pow(fac[x], mod - 2, mod) % mod
        for x in nums[1:-1]:
            if x > 1:
                ans = ans * pow(2, x - 1, mod) % mod
        return ans
