# Choosing $$n$$ candies from $$m$$ brands

In how many ways can we choose $$n$$ candies from $$m$$ brands?

Note: Repeated selection from the same brand is allowed and $$n\leq m$$.

Why is $$m^n$$ not the correct answer?

Note by D K
2 years, 2 months ago

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If $$n$$ candies are chosen from $$m$$ brands, then the sum of the number of candies from each brand equals $$n$$.(That's pretty obvious right?!).Suppose $$x_1$$ candies are chosen from Brand #1 , $$x_2$$ candies from Brand #2 and ... $$x_m$$ candies from Brand #$$m$$.Now continuing my argument above , obviously the answer to your question is equivalent to the number of answers to the equation :

$$\space$$

$$x_1 + x_2 + \dots + x_m = m$$ , $$x_i \ge 0$$ ;

$$\space$$

Now if you're familiar with "Stars and bars" you'd know that the answer is $$n+m-1 \choose m-1$$.(If not , you can read it's wikipage here , I'm too lazy to write the whole thing down here)

Now as for the answer to your second question,the answer $$m^n$$ would definitely not be correct since you're not counting the cases where no candies are chosen from a particular brand also you're not talking into account the fact that candies from different brands are not alike.I hope I could help you get a good grasp on this.

- 2 years, 2 months ago