# How many ways are there? [closed]

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?

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

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• 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 $$x_1 + x_2 + x_3 = 14-8$$ – Matti P. Dec 11 '18 at 7:04
• I think you have to multiply above result by 3 ? @MattiP. – rsadhvika Dec 11 '18 at 7:14
• @rsadhvika I agree. Multiplying by 3 takes into account the 8 objects in the beginning. – Matti P. Dec 11 '18 at 7:15
• I suggest looking up "generating functions" as a way to solve these integer problems. – Aditya Dua Dec 11 '18 at 7:17

## 1 Answer

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?

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