<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:
    def countOrders(self, n: int) -> int:
        # first time, can only choose one out of any pickup
        # next time, can choose one out of any pickup - 1 or 1 delivery
        # next time, can choose one out of any pickup - 2 or 2 delivery, 

        MOD = 10**9 + 7
        
        @functools.cache
        def count_options(available_to_pick, available_to_deliver):
            # print("count_options", available_to_pick, available_to_deliver)
            if not available_to_pick and not available_to_deliver:
                return 1
            options_if_picked = 0 if not available_to_pick else available_to_pick * count_options(available_to_pick - 1, available_to_deliver + 1) % MOD
            options_if_delivered = 0 if not available_to_deliver else available_to_deliver * count_options(available_to_pick, available_to_deliver - 1) % MOD
            # print("options", options_if_picked, options_if_delivered)
            # Now try picking if possible and delivering if possible.
            return options_if_picked + options_if_delivered % MOD
        
        return count_options(n, 0)

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