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In how many ways can 15 indistinguishable fishes be placed into 5 different ponds, so that each pond contains atleast one fish?

I am struck on this problem.Can someone help me out please.

manu
  • 43
  • 4

3 Answers3

2

Arrange those $15$ indistinguishable fishes in a row.

Consider those $5$ ponds as $4$ delimiters, and place them anywhere between the fishes, so that:

  • No delimiter is at the beginning of the row or at the end of the row
  • No two delimiters are next to each other (i.e., without fishes in between them)

So there are essentially $14$ places to choose without replacement for these $4$ delimiters.

Hence there are $\binom{14}{4}=1001$ ways to do it.

barak manos
  • 43,109
2

Firstly, put a fish each in each pond,

we now need to find how many ways to place the remaining 10 such that $x_1+x_2+x_3+x_4+x_5=10,\; x_i\ge0$

Consider a series of fishes totalling to 10 separated by $+'s$, e.g. $FFF+ FFFF + F + F +FF$

The only thing we need to decide is where to place the $+'s$ among the 14 symbols,

$$\text{thus answer} = {14\choose4} = 1001$$

true blue anil
  • 41,295
0

Here there is a VB program to be run in Excel, showing the 1001 forms to put 15 fishes in five ponds:

Sub Macro1418025()

CONT = 1

For I = 1 To 15
For J = 1 To 15
For K = 1 To 15
For L = 1 To 15
For M = 1 To 15

Sum = I + J + K + L + M
If Sum = 15 Then
Cells(CONT, 1) = I
Cells(CONT, 2) = J
Cells(CONT, 3) = K
Cells(CONT, 4) = L
Cells(CONT, 5) = M
CONT = CONT + 1
End If

Next M
Next L
Next K
Next J
Next I

End Sub
cgiovanardi
  • 836