How many different ways can 12 pieces of candy be put into 3 jars?

This problem involves combinations with repetition. Here's how to solve it:

Understanding the Problem

Imagine the candy pieces are identical (like 12 gumballs). We want to find how many different ways we can distribute them into the three jars, where the order of the jars doesn't matter (jar 1 having 5 and jar 2 having 4 is the same as jar 2 having 4 and jar 1 having 5).

The Formula

The number of ways to distribute *n* identical items into *k* distinct containers is:

(n + k - 1) choose (k - 1) = (n + k - 1)! / (k - 1)! * n!

Applying the Formula

* n = 12 (number of candy pieces)

* k = 3 (number of jars)

So, the calculation becomes:

(12 + 3 - 1) choose (3 - 1) = 14 choose 2 = 14! / (2! * 12!) = 91

Answer: There are 91 different ways to put 12 pieces of candy into 3 jars.