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Can we generalize it?

I was working on one question:- if there are 1 red balls, 2 blue balls, 3 green balls, 4 yellow balls and 5 white balls placed on a table, what is the number of ways of selecting 5 out of 5. My process came out with 7 cases as follows:-

  • 5 alike balls.
  • 4 alike balls and 1 different ball.
  • 3 alike balls and 2 different balls.
  • 3 alike balls of one colour and 2 alike balls of another colour.
  • 2 alike balls and 3 different balls.
  • 2 alike balls of one colour, 2 alike balls of another colour and 1 different ball.
  • 5 different balls.

I got interested and tried to generalize the number of such cases that would form if we have to select \(n\) balls from 1 ball of colour-1, 2 balls of colour-2, 3 balls of colour-3, \(\cdots n\) balls of colour \(n\).

It turns out that when \(n = 1\), the number of cases is \(1\).
When \(n = 2\), the number of cases is \(2\).
When \(n = 3\), the number of cases is \(3\).
When \(n = 4\), the number of cases is \(5\).
When \(n = 5\), the number of cases is \(7\). (as discussed above).
When \(n = 6\), the number of cases is \(12\).

Now, just in case that the sixth case becomes clear, here are the cases I found:-

  • 6 alike.
  • 5 alike and 1 different.
  • 4 alike and 2 different.
  • 4 alike and 2 alike.
  • 3 alike and 3 different.
  • 3 alike and 2 alike and 1 different.
  • 3 alike and 3 alike.
  • 2 alike and 4 different.
  • 2 alike and 3 alike and 1 different.
  • 2 alike and 2 alike and 2 different.
  • 2 alike and 2 alike and 2 alike.
  • 6 different.

Looking at the pattern, the first thing which came into my mind is the fibonacci sequence, but as you can observe, the pattern is broken at \(n = 5\). So, is there any nice method to calculate the number of cases that can be formed?

A detailed solution is appreciated.

Note by Ashish Siva
4 months ago

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@Hung Woei Neoh @Geoff Pilling @Sambhrant Sachan @Sandeep Bhardwaj plz do comment. Special thanks in advance! Ashish Siva · 4 months ago

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@Ashish Siva Your problem is interesting Sambhrant Sachan · 4 months ago

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@Sambhrant Sachan Thanks! Its 1ould great tp know the number of cases that would be rormed before going through making the exhaustive cases, so that we can be sure of our answer. My teacher replied that he needs time to think over it. Ashish Siva · 4 months ago

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Very interesting... Lemme think about it for a bit and see what I can come up with... Geoff Pilling · 4 months ago

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@Geoff Pilling Sure, take your time. Please notify me if you come up with something. Thanks again! :) Ashish Siva · 4 months ago

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