<p>Given <code>n</code> orders, each order consists of a pickup and a delivery service.</p>

<p>Count all valid pickup/delivery possible sequences such that delivery(i) is always after of&nbsp;pickup(i).&nbsp;</p>

<p>Since the answer&nbsp;may be too large,&nbsp;return it modulo&nbsp;10^9 + 7.</p>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> Unique order (P1, D1), Delivery 1 always is after of Pickup 1.
</pre>

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

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 6
<strong>Explanation:</strong> All possible orders: 
(P1,P2,D1,D2), (P1,P2,D2,D1), (P1,D1,P2,D2), (P2,P1,D1,D2), (P2,P1,D2,D1) and (P2,D2,P1,D1).
This is an invalid order (P1,D2,P2,D1) because Pickup 2 is after of Delivery 2.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> 90
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 500</code></li>
</ul>

class Solution:

    @functools.cache
    def totalWays(self, unpicked, undelivered):
        MOD = 1_000_000_007
        if not unpicked and not undelivered:
            # We have completed all orders.
            return 1

        if (unpicked < 0 or undelivered < 0 or undelivered < unpicked):
            # We can't pick or deliver more than N items
            # Number of deliveries can't exceed number of pickups 
            # as we can only deliver after a pickup.
            return 0

        # Count all choices of picking up an order.
        ans = unpicked * self.totalWays(unpicked - 1, undelivered)
        ans %= MOD

        # Count all choices of delivering a picked order.
        ans += (undelivered - unpicked) * self.totalWays(unpicked, undelivered - 1)
        ans %= MOD

        return ans

    def countOrders(self, n: int) -> int:
        return self.totalWays(n, n)

        # MOD = 1_000_000_007
        # ans = 1

        # for i in range(1, n + 1):
        #     # Ways to arrange all pickups, 1*2*3*4*5*...*n
        #     ans = ans * i
        #     # Ways to arrange all deliveries, 1*3*5*...*(2n-1)
        #     ans = ans * (2 * i - 1)
        #     ans %= MOD
        
        # return ans
189400