<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>k</code>.</p>

<p>You have an infinite number of coins of each denomination. However, you are <strong>not allowed</strong> to combine coins of different denominations.</p>

<p>Return the <code>k<sup>th</sup></code> <strong>smallest</strong> amount that can be made using these coins.</p>

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

<div class="example-block" style="
    border-color: var(--border-tertiary);
    border-left-width: 2px;
    color: var(--text-secondary);
    font-size: .875rem;
    margin-bottom: 1rem;
    margin-top: 1rem;
    overflow: visible;
    padding-left: 1rem;
">
<p><strong>Input:</strong> <span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
">coins = [3,6,9], k = 3</span></p>

<p><strong>Output:</strong> <span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> 9</span></p>

<p><strong>Explanation:</strong> The given coins can make the following amounts:<br />
Coin 3 produces multiples of 3: 3, 6, 9, 12, 15, etc.<br />
Coin 6 produces multiples of 6: 6, 12, 18, 24, etc.<br />
Coin 9 produces multiples of 9: 9, 18, 27, 36, etc.<br />
All of the coins combined produce: 3, 6, <u><strong>9</strong></u>, 12, 15, etc.</p>
</div>

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

<div class="example-block" style="
    border-color: var(--border-tertiary);
    border-left-width: 2px;
    color: var(--text-secondary);
    font-size: .875rem;
    margin-bottom: 1rem;
    margin-top: 1rem;
    overflow: visible;
    padding-left: 1rem;
">
<p><strong>Input:</strong><span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> coins = [5,2], k = 7</span></p>

<p><strong>Output:</strong><span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> 12 </span></p>

<p><strong>Explanation:</strong> The given coins can make the following amounts:<br />
Coin 5 produces multiples of 5: 5, 10, 15, 20, etc.<br />
Coin 2 produces multiples of 2: 2, 4, 6, 8, 10, 12, etc.<br />
All of the coins combined produce: 2, 4, 5, 6, 8, 10, <u><strong>12</strong></u>, 14, 15, etc.</p>
</div>

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

<ul>
	<li><code>1 &lt;= coins.length &lt;= 15</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 25</code></li>
	<li><code>1 &lt;= k &lt;= 2 * 10<sup>9</sup></code></li>
	<li><code>coins</code> contains pairwise distinct integers.</li>
</ul>

class Solution:
    def findKthSmallest(self, coins: List[int], k: int) -> int:
        
        n = len(coins)
        l, r = 0, 10 ** 18

        def isLargerOrEqual(target):
            nonlocal coins, k 

            count = 0 
            for mask in range(0, 1<<n) :

                lcm = 0 
                sub_coins = []
                for i in range(0, n):
                    if (mask >> i) & 1 == 1:
                        sub_coins.append(coins[i])
                
                lcm = math.lcm(*sub_coins)

                if lcm != 0 and len(sub_coins) != 0:
                    if len(sub_coins) % 2 == 1:
                        count += target // lcm
                    elif len(sub_coins) % 2 == 0: 
                        count -= target // lcm
            return k <= count

        while l+1 < r:
            m = (l + r) >> 1; 
            if isLargerOrEqual(m) :
                r = m
            else : 
                l = m 

        return r
1866