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

  1. $1001$
  2. $3876$
  3. $775$
  4. $200$
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$x_{1} + x_{2}+ \dots+ x_{n}= r\;, x_{i}\geq 0$.          The number of solutions $=\binom{n+r-1}{r}$
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This given question is equivalent to distribution of 15 identical objects into 5 distinct boxes where each box contains at least one object.

Number of  ways it can be done = number of positive integer solutions of

=>  x1+x2+x3+x4+x5=15x1+x2+x3+x4+x5=15  
=> (15-1)C4 = 14C4 = 1001

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given that distribution of 15 identical fish into 5 distinct ponds where each box contains at least one fish. 

Number of ways fishes placed in ponds

= $\binom{n-1}{r-1}$=$\binom{14}{4}$ =1001

Hence.Option(A)1001.

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This is simple IODB template where 15 fishes are stars and 5 ponds means 4 bars.

 

Therefore 15stars and a+b+c+d+e=15stars.

 

Now you directly give 1 fish to each pond

 

THen the equation becomes a+b+c+d+e=10

 

Now there are 10 stars and 4 bars.-→ (14)C4===1001
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