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April 26th, 2014, 03:34 AM   #1
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place balls in boxes

Hello!!!!

With how many ways can we place n similar balls in k boxes,which are numbered with the numbers 1,...,k?
The same question,when the balls are also numbered.
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April 26th, 2014, 08:15 AM   #2
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Math Focus: सामान्य गणित
1 way
n! / (n - k)! ways
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April 26th, 2014, 08:19 AM   #3
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Quote:
Originally Posted by MATHEMATICIAN View Post
1 way
n! / (n - k)! ways
How did you find it?
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April 26th, 2014, 08:50 AM   #4
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sorry i misunderstood your question.

first case :
ways : k! / {(k-n)!n!}

second case :
ways : k! / (k - n)!
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April 26th, 2014, 09:03 AM   #5
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Quote:
Originally Posted by MATHEMATICIAN View Post
sorry i misunderstood your question.

first case :
ways : k! / {(k-n)!n!}

second case :
ways : k! / (k - n)!
But how did you find it? I still haven't understood it.
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April 26th, 2014, 09:23 AM   #6
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Math Focus: सामान्य गणित
you have k empty boxex and n balls ( n < k )

for the first ball : you can place it in any one of k boxes
for the second ball : you can place it in any one of k-1 boxes (as one box is filled with first ball)
for the third ball : you can place it in any one of k-2 boxes (as two boxes is filled with first and second balls)
.
.
.
.
finally, for the nth ball : you can place it in any one of k-(n-1) boxes (as (n-1) boxes is filled with (n-1) balls)

then total number of ways to fill up the boxes
= k(k-1)(k-2) . . . . . (k-(n-1))
multiplying and dividing by (k-n)!
= k(k-1)(k-2) . . . . . (k-(n-1))(k-n)! / (k-n)!
= k! / (k-n) !

this gives number of arrangement for numbered balls
if the balls are similar, the position of balls with respect to each other would not matter
the balls can be arranged among themselves in n! ways

so for similar balls :
number of arrangement = k! / {(k-n)!n!}

Last edited by MATHEMATICIAN; April 26th, 2014 at 09:30 AM.
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April 26th, 2014, 02:58 PM   #7
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Hello, evinda!

Quote:
(a) In how many ways can we place $\displaystyle n$ similar balls in $\displaystyle k$ boxes,
which are numbered with the numbers $\displaystyle 1,2,3,\cdots,k$?

(b) The same question, when the balls are also numbered.

I will assume that some boxes may be empty.

(a) $\displaystyle n$ identical balls and $\displaystyle k$ numbered boxes.

I'll illustrate with a specific example.
We have: $\displaystyle \,n = 7$ balls and $\displaystyle k = 3$ boxes.

Place the 7 balls in a row, inserting a space before, after and between them.
$\displaystyle \qquad \_\,\circ\,\_\,\circ\,\_\,\circ\,\_\,\circ\,\_ \, \circ\,\_\,\circ\,\_\,\circ\,\_$

Distribute 2 "dividers" among the 8 spaces.

$\displaystyle \quad \circ\,\circ\,|\,\circ\,|\,\circ\,\circ\,\circ\, \circ$ represents $\displaystyle (2,1,4).$

$\displaystyle \quad \circ\,\circ\,\circ\,\circ\,\circ\,|\,|\,\circ\, \circ$ represents $\displaystyle (5,0,2).$

$\displaystyle \quad |\,|\,\circ\,\circ\,\circ\,\circ\,\circ\,\circ\, \circ$ represents $\displaystyle (0,0,7).$

There are: $\displaystyle \,8^2 \:=\:64$ ways.


In general, the number is: $\displaystyle \,(n+1)^{k-1}$

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April 27th, 2014, 12:43 AM   #8
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i misunderstood second time also
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