Suppose to have a jar containing 100 coins. I want to count the possibile configuration with pennies, nickels, dimes, quarters and half-dollars.
This is what I have done, but I realized that it's wrong: we have 5 type of labels and we want to label the coins with $k_1,\ldots,k_5$ labels such that $k_1+\cdots+k_5=100$, so what I want to calculate is $\sum_{k_i\geq0,\sum k_i=100}\binom{100}{k_1,\ldots,k_5}$, I calculated that this number is $5^{100}$ (well it's obvious that it's $5^{100}$). But I realized that I'm counting too many objects. Do you have any idea to solve this problem?