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How many ways are there to put 14 identical objects in 3 distinct boxes with at least 8 objects in one box?

What is the thinking procedure of the similar question type?

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closed as off-topic by choco_addicted, Henrik, Cesareo, GNUSupporter 8964民主女神 地下教會, José Carlos Santos Dec 11 '18 at 11:52

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    $\begingroup$ If you don't regard the condition with 8 objects in the beginning, the problem is similar to finding the number of integer solutions to $$ x_1 + x_2 + x_3 = 14-8 $$ $\endgroup$ – Matti P. Dec 11 '18 at 7:04
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    $\begingroup$ I think you have to multiply above result by 3 ? @MattiP. $\endgroup$ – rsadhvika Dec 11 '18 at 7:14
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    $\begingroup$ @rsadhvika I agree. Multiplying by 3 takes into account the 8 objects in the beginning. $\endgroup$ – Matti P. Dec 11 '18 at 7:15
  • $\begingroup$ I suggest looking up "generating functions" as a way to solve these integer problems. $\endgroup$ – Aditya Dua Dec 11 '18 at 7:17
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You have to find a unique way to describe a possible placement of the objects, that makes counting easy. Here is a solution.

  • Choose the box that has at least $8$ objects: $3$ choices.
  • For each of the preceding choices, let $k$ be the number of objects in this box ($8\le k\le 14$).
  • For each $k$, choose the number of objects in the leftmost remaining box ($0$ to $14-k$): $15-k$ choices.
  • The last box gets the remaining.

This uniquely describes the three boxes.

How many choices now?

$$3\sum_{k=8}^{14} (15-k)$$

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