GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 26

Question

In a college $10$ Servers are bought and they are to be distributed among $4$ laboratories such that each laboratory gets at least one server. The total number of ways in which this can be done is $\_\_\_$

Answer 1

This is a balls-in-bins problem with balls being identical and no bins being empty. This problem can be reduced to the integer sum problem

$x_1+x_2+x_3+x_4 = 10, x_i > 0$

This is same as

$x_1+x_2+x_3+x_4 = 6, x_i \geq 0$
(we allotted $1$ server to each laboratory)

Now, the above problem can be solved by putting $3$ separations among $6$ identical objects and then considering all the permutations as follows:

$******|||$

The three $|$ represent the $4$ regions corresponding to the $4$ bins. In the above representation (stars-and-bars), first bin is having the $6$ balls and the rest of the bins are empty. Now, the total number of permutations of the above (total $9$ objects where $6$ are of one kind and $3$ are of other) is
$\frac{9!}{6!.3!} = 84.$

Comments

@ketandalvi we can do it if we consider servers distinct, but here since we are considering servers identical, we are using balls in bins approach.

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@Yogesh88 So , why in NALANDA PROPERTY question they have not mentioned distinct ??

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@ GATE2021

In that Qn it is mentioned “distinct property” and daughters are also (obviously) distinct.