Sample space, indistinguishables and distinguishables balls

Question

I'm working on balls and samples spaces.

What is the sample space of 2 indistinguishable balls in 4 cells? and

What is the sample space of 2 distinguishable balls in 4 cells?

Thanks!.

Answer 1

What is the sample space of 2 indistinguishable balls in 4 cells? and

Using the string "$\star\star\circ\star\circ$" to denote placing a ball in the third and fourth cells, and such, we have the set of all arrangements of two balls and three cell-dividers.

$\{\circ\circ\star\star\star, \circ\star\circ\star\star, \circ\star\star\circ\star, \ldots, \star\star\star\circ\circ\}$

There are $\dfrac{5!}{2!~3!}$ elements in this set, which is unsurprisingly the forth triangle number $\frac{4(4+1)}{2}$

This set has a bijection with $\{(1,1), (1,2), (1,3), (1,4), (2,2), \ldots (4,4)\}$ or $\{(x,y)\in \Bbb N^2: 1\leq x\leq y\leq 4\}$ , the set of selected cell numbers to place balls, where order is not important but repetition is allowed.

What is the sample space of 2 distinguishable balls in 4 cells?

We use a similar notation.

$\{\circ\bullet\star\star\star, \circ\star\bullet\star\star, \bullet\star\circ\star\star, \circ\star\star\bullet\star, \bullet\star\star\circ\star, \ldots, \star\star\star\circ\bullet\}$

A note should be made that when the balls are in the same cell, their order is irrelevant, and so we do not include the four arrangements such as "$\star\bullet\circ\star\star$" .

There are thus $\dfrac{5!}{3!}-4$ arrangements in the set which, again unsurprisingly, equals $4^2$.

This sample space has a bijection with $\{(1,1), (1,2), (2,1),\ldots, (4,4)\}$, of $\{(x,y)\in\Bbb N^2: 1\leq x\leq 4, 1\leq y\leq 4\}$, the set of ordered pairs of cell numbers for each ball.