The number of ways by which 6 distinct balls can be put in 5 distinct boxes are

{'distinct balls': [0, 0, 0, 1], 'distinct boxes': [0, 0, 0, 1]}

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The number of ways by which 6 distinct balls can be put in 5 distinct boxes are

The number of ways by which 6 distinct balls can be put in 5 distinct boxes are
The number of ways by which 6 distinct balls can be put in 5 distinct boxes are

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How many ways are there to distribute six indistinguishable balls into nine distinguishable bins?

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My approach :-

I first assumed all balls to be similar in nature , so that would give me 5 ways to distribute the balls in the boxes , which will be

2 1 1 1 1

1 2 1 1 1

1 1 2 1 1

1 1 1 2 1

1 1 1 1 2

basically that would be whole number solutions of a+b+c+d+e = 6 where all a,b,c,d,e >=1

Now since all the balls are distinct in nature I multiplied the 5 ways with 6! = giving me a total of 3600 ways , but the answer is given as 1800 ways , where am I going wrong ?

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The number of ways by which 6 distinct balls can be put in 5 distinct boxes are

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Given:

6 distinct balls can be put in 5 distinct boxes.

Calculation:

The number of ways of in n distinct objects can be put into identical boxes, so that neither one of them remains empty.

Since both the boxes and the balls are different , we can choose any box , and every choice is different at any time.

The first ball can be placed in any of the 5 boxes . Similarly , the other balls can be placed in any of the 5 boxes.

The number of ways = 56 = 15625

∴ The number of ways is 15625.

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