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I'm self studying discrete math from a books which states the formula for No of ways of selecting r objects from n distinct objects, allowing repeated selections as $C(n+r-1, r)$. I couldn't understand completely how they derived this formula. Could someone please explain this to me?

Kindly also do give some examples to help me understand how to use this formula. The book is elements of discrete mathematics by KC liu and DP mohapatra


marked as duplicate by 5xum, David K, user147263, user98602, Micah Jul 7 '15 at 20:16

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    $\begingroup$ Kind of hard to explain how they derived the formula when you don't tell us the book title or cite how they claim to derive it... $\endgroup$ – 5xum Jul 7 '15 at 12:31
  • $\begingroup$ Since there is no indication of what the book said about this problem, other than the formula, I presume the question is "explain why the formula is $C(n+r-1,r)$" rather than "explain the proof in the book." That is a question that has been asked and answered before. $\endgroup$ – David K Jul 7 '15 at 13:56

This formula can be reached quite easily. Think of this problem as n choices of ice cream and r scoops. Put the ice cream choices in a row. If you start from one end and move across, you will have a total of n-1 moves and r scoops, which gets the n+r-1 part of the formula. Then, from these, you have to choose r scoops, which gets you the formula, C(n+r−1,r). Also, this website https://www.mathsisfun.com/combinatorics/combinations-permutations.html may help you to understand.

  • $\begingroup$ I still didn't get r in n+r-1 part $\endgroup$ – Arpit Quickgun Arora Jul 7 '15 at 12:43

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