GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 18

Answer 1

I made a mistake in reading the questions actually piles should be non-empty but I missed that point but no problem, I realized my mistake.

now comes the solution part it is the IOIB template so we have to divide 10 similar  objects into 3 similar boxes,

so we can find it manually3

 

A	B	C
1	1	8
1	2	7
1	3	6
1	4	5
2	3	5
2	2	6
2	4	4
3	3	4

In the above table all A, B, and C are indistinguishable boxes, and the number in their respective columns is the number of objects(indistinguishable).

now in the IOIB template order doesn’t matter as 1 1 8 and 1 8 1 are the same things 

Comments

@MunchyMoo Then answer will be 14 these 8 ways in 3 piles and 5 ways in 2 piles and 1 way in 1 pile.

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@Swarnava Bose There might be a formula I don't know about it, but it is the most efficient way to solve. Thank you for identifying the mistake

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@svmkumar28 https://gateoverflow.in/421287/%23self-doubt here it is the explanation for 9330.

Answer 2

If both the objects and the piles are indistinguishable, a division is characterized solely by the number of objects in a pile.

We can easily list these different possibilities systematically:
$$
\begin{aligned}
8-1-1\\7-2-1\\6-3-1\\5-4-1\\4-4-2 \\ 6-2-2\\5-3-2\\4-3-3
\end{aligned}
$$
Recall that the piles should not be empty. So we can do this in eight ways.

Comments

instead of doing this manually, is there any shortcut or formula ?