To kick of this school year, the following image shows my worked out answers to the MATHCOUNTS Problem of the Week published on 8/14/2017. When the official answers and solutions are posted, this post will be updated to see how I did. By the way, this is an example of an enrichment activity that I do with middle school students when time allows during a tutoring session.
Here is the link to the published solutions for the 8/14/2017 MATHCOUNTS Problem of the Week. My answers to the first and third problems were correct, while my answer to the second problem was incorrect. Here's why. There is only one way to pick at least one of each of the four colors, and not 24 ways (4! = 4 x 3 x 2 x 1) as I originally thought. The problem then becomes the number of ways to choose from four colors for each of the two remaining notebooks. Next, there are only four ways to choose two of the same color. What remains is the number of two color combinations. In other words, the number of ways to choose two from four items:
4C2 = 4! / (2! x 2!) = 4x3x2x1 / 2x1x2x1 = 24/4 = 6
Finally, add 4 and 6 to get the final answer of 10 for the second question.
Mr. Ken Marciel