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POMaximum Possible Ways of Merging Two Arrays
 

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  1. POMaximum Possible Ways of Merging Two Arrays
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    <p>Suppose I have two arrays of size <code>m</code> and <code>n</code>:</p> <p><code>a[1] a[2] a[3] ..... a[m]</code></p> <p>and</p> <p><code>b[1] b[2] b[3] ..... b[n]</code></p> <p>I want to form a new array merging these two arrays such that in the new array of <code>m + n</code> elements, <code>a[i]</code> is always placed befor <code>a[i + 1]</code> and <code>b[i]</code> is always placed before <code>b[i + 1]</code>. For example, <code>a[1] a[2] b[1] b[2]... b[n] a[m]</code> will be a valid array but <code>a[2] a[1] b[1] b[2] ... b[n] a[m]</code> won't. Given <code>m</code> and <code>n</code>, how many such combinations will be possible when repeating is allowed?</p> <p>I have the intuition to solve the problem:</p> <p><code>- b[1] - b[2] - b[3] - ..... - b[n]</code></p> <p>I can place <code>a[1]</code> in any of the <code>n - 1</code> places within the array <code>b</code>, and considering the front and the last place, I have <code>n + 1</code> total ways of placing <code>a[1]</code>. If I place <code>a[1]</code> in the first place (just before <code>b[1]</code>), I can now place <code>a[2]</code> in <code>n + 1</code> places. But if I place <code>a[1]</code> just after <code>b[1]</code>, I would have <code>n</code> ways to place <code>a[2]</code>. I can apply this approach recursively for all <code>a[i]</code> where <code>1 &lt;=i &lt;= n</code>. But I can't find any mathematical formula to express the solution, besides I can't understand how to approach when repeating is allowed.</p>
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    1. USRafi Kamal
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    1. COYou should try math.stackexchange.com, or maybe cstheory
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    2. COThis is called an _interleaving_, see here: http://math.stackexchange.com/questions/6801/how-do-i-determine-the-possible-number-of-combinations-of-two-ordered-sets
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      1. POMaximum Possible Ways of Merging Two Arrays
      1. USAndrew Tomazos
 

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