build-array-where-you-can-find-the-maximum-exactly-k-comparisons

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Algorithms

You are given three integers <code>n</code>, <code>m</code> and <code>k</code>. Consider the following algorithm to find the maximum element of an array of positive integers:
<img alt="" src="https://assets.leetcode.com/uploads/2020/04/02/e.png" style="width: 424px; height: 372px;" />
You should build the array arr which has the following properties:

<ul>
	<li><code>arr</code> has exactly <code>n</code> integers.</li>
	<li><code>1 &lt;= arr[i] &lt;= m</code> where <code>(0 &lt;= i &lt; n)</code>.</li>
	<li>After applying the mentioned algorithm to <code>arr</code>, the value <code>search_cost</code> is equal to <code>k</code>.</li>
</ul>

Return the number of ways to build the array <code>arr</code> under the mentioned conditions. As the answer may grow large, the answer must be computed modulo <code>109 + 7</code>.

&nbsp;
Example 1:

<pre>
Input: n = 2, m = 3, k = 1
Output: 6
Explanation: The possible arrays are [1, 1], [2, 1], [2, 2], [3, 1], [3, 2] [3, 3]
</pre>

Example 2:

<pre>
Input: n = 5, m = 2, k = 3
Output: 0
Explanation: There are no possible arrays that satisfy the mentioned conditions.
</pre>

Example 3:

<pre>
Input: n = 9, m = 1, k = 1
Output: 1
Explanation: The only possible array is [1, 1, 1, 1, 1, 1, 1, 1, 1]
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= n &lt;= 50</code></li>
	<li><code>1 &lt;= m &lt;= 100</code></li>
	<li><code>0 &lt;= k &lt;= n</code></li>
</ul>

numOfArrays

Build Array Where You Can Find The Maximum Exactly K Comparisons

Dynamic Programming

Prefix Sum

给定三个整数 <code>n</code>、<code>m</code> 和 <code>k</code> 。考虑使用下图描述的算法找出正整数数组中最大的元素。

<img alt="" src="https://assets.leetcode-cn.com/aliyun-lc-upload/uploads/2020/04/19/e.png" style="height: 372px; width: 424px;" />

请你构建一个具有以下属性的数组 <code>arr</code> ：

<ul>
	<li><code>arr</code> 中包含确切的&nbsp;<code>n</code> 个整数。</li>
	<li><code>1 &lt;= arr[i] &lt;= m</code> 其中 <code>(0 &lt;= i &lt; n)</code> 。</li>
	<li>将上面提到的算法应用于 <code>arr</code>&nbsp;之后，<code>search_cost</code> 的值等于 <code>k</code> 。</li>
</ul>

返回在满足上述条件的情况下构建数组 <code>arr</code> 的 方法数量&nbsp;，由于答案可能会很大，所以 必须 对 <code>10^9 + 7</code> 取余。

&nbsp;

示例 1：

<pre>
输入：n = 2, m = 3, k = 1
输出：6
解释：可能的数组分别为 [1, 1], [2, 1], [2, 2], [3, 1], [3, 2] [3, 3]
</pre>

示例 2：

<pre>
输入：n = 5, m = 2, k = 3
输出：0
解释：没有数组可以满足上述条件
</pre>

示例 3：

<pre>
输入：n = 9, m = 1, k = 1
输出：1
解释：唯一可能的数组是 [1, 1, 1, 1, 1, 1, 1, 1, 1]
</pre>

&nbsp;

提示：

<ul>
	<li><code>1 &lt;= n &lt;= 50</code></li>
	<li><code>1 &lt;= m &lt;= 100</code></li>
	<li><code>0 &lt;= k &lt;= n</code></li>
</ul>