If the balls and boxes are both distinguishable, then for each ball, there are $8$ choices of which box to place it in, for a total of $8 \times 8 \times 8 \times 8 = \boxed{4096}$ ways.
If neither the balls nor the boxes are distinguishable, then we are counting the number of ways to divide $4$ indistinguishable balls into $8$ indistinguishable boxes. This is equivalent to the number of nonnegative solutions to the equation $x_1 + x_2 + x_3 + \dots + x_8 = 4$. We can think of this as placing $4$ dots in a row and then placing $7$ dividers between them to divide them into $8$ boxes. For example, if we have the arrangement $\cdot \cdot \cdot | \cdot | \cdot \cdot | \cdot \cdot$, then there are $3$ balls in the first box, $1$ ball in the second box, $2$ balls in the third box, and $0$ balls in the remaining boxes. There are a total of $11$ objects ($4$ balls and $7$ dividers), and we must choose $4$ of them to be balls, so there are $\binom{11}{4} = \boxed{330}$ ways.
If the balls are indistinguishable and the boxes are distinguishable, then we are counting the number of ways to divide $4$ indistinguishable balls into $8$ distinguishable boxes. This is equivalent to the number of nonnegative solutions to the equation $x_1 + x_2 + x_3 + \dots + x_8 = 4$. Again, we can think of this as placing $4$ dots in a row and then placing $7$ dividers between them to divide them into $8$ boxes. Using the stars and bars formula, there are $\binom{4+8-1}{8-1} = \binom{11}{7} = \boxed{330}$ ways.
If the balls are distinguishable and the boxes are indistinguishable, then there are $8$ choices of which box to place the first ball in, $8$ choices of which box to place the second ball in, and so on, for a total of $8 \times 8 \times 8 \times 8 = \boxed{4096}$ ways.
In how many ways can $4$ balls be placed in $8$ boxes if the balls and boxes are both distinguishable?
In how many ways can $4$ balls be placed in $8$ boxes if neither the balls nor the boxes are distinguishable?
In how many ways can $4$ balls be placed in $8$ boxes if the balls are indistinguishable, and the boxes are distinguishable?
In how many ways can $4$ balls be placed in $8$ boxes if the balls are distinguishable, and the boxes are indistinguishable?
1 answer