edited by
7,062 views

2 Answers

6 votes
6 votes

$8$ indistinguishable(identical) balls into $4$ distinguishable(distinct) bins.

There are could be empty bins.

 


Using Stars and Bars, we can correlate the indistinguishable(identical) balls into stars and  $4$ distinguishable(distinct) bins into bars.

$8$  indistinguishable(identical) balls like $8$ stars ${\color{Green}{  \bigstar \ \bigstar \ \bigstar \ \bigstar \ \bigstar \ \bigstar \ \bigstar \ \bigstar \ \ } }$

separate $8$ stars into $4$ distinguishable(distinct) bins we can do like this 

 ${\color{Green}{\bigstar}} \ {\color{Blue}{\mid}} \ {\color{Green}{\bigstar \bigstar}} \ {\color{Magenta}{\mid}} \ {\color{Green}{\bigstar\bigstar}} \ {\color{DarkOrange}{\mid}} \ {\color{Green}{\bigstar\bigstar\bigstar}}$ so we need three bars.

There are $8 + 3 = 11$ things, that needs to be placed and $3$ of those placements are chosen for the bars.

Thus, there are $\binom{11}{3} = 165$ possible distribution.

  • Suppose there are $n$ identical objects to be distributed among $r$ distinct bins. This can be done in precisely $\binom{n+r-1}{r-1}$ ways.

References:

edited by
5 votes
5 votes

Correct answer is option 2. 


Detailed explanation is provided in the pic below:


For case b in the picture above i.e. number of balls m <= number of bags n , it is assumed that one bag cannot contain more than one ball. In that case number of ways will be nCm.

But if the question says that a bag can contain any number of balls, then number of ways will become (n+m-1)Cm.

It must be mentioned in the question. According to the question it must be decided.

 

edited by
Answer:

Related questions

3 votes
3 votes
2 answers
1
Arjun asked Jul 2, 2019
5,750 views
How many bit strings of length ten either start with a $1$ bit or end with two bits $00$ ?$320$$480$$640$$768$
4 votes
4 votes
1 answer
2
Arjun asked Jul 2, 2019
5,960 views
How many cards must be selected from a standard deck of $52$ cards to guarantee that at least three hearts are present among them?$9$$13$$17$$42$
2 votes
2 votes
1 answer
3
Arjun asked Jul 2, 2019
2,752 views
Consider the Euler’s phi function given by$$\phi(n) = n \underset{p/n}{\Pi } \bigg( 1 – \frac{1}{p} \bigg)$$where $p$ runs over all the primes dividing $n$. What is t...