Julie baked cupcakes for her family at home and for a party at school. She iced 4 cupcakes with red frosting, 2 cupcakes with orange frosting, 2 with yellow, 2 with green, 3 with blue and the last 3 with violet frosting. Each cupcake is iced with exactly one color of frosting. Julie plans to take exactly 10 of the cupcakes to her party, and will take either all of the cupcakes of a particular color or none of the cupcakes of that color. How many different combinations of cupcakes could she take to her party?

Well... to start with "order doesn't matter" so you are dealing with a combination.

Please state the combination expression you believe to be the make up of your solution. A tutor will be happy to review your expression.
PS... what a tasty example : )

To find the number of different combinations of cupcakes Julie could take to her party, we need to consider the possible options for each color of frosting.

For the red frosting, she can either take all 4 cupcakes or none of them (0 cupcakes).
For the orange frosting, she can take both cupcakes or none of them (0 cupcakes).
For the yellow frosting, she can take both cupcakes or none of them (0 cupcakes).
For the green frosting, she can take both cupcakes or none of them (0 cupcakes).
For the blue frosting, she can take all 3 cupcakes, 2 cupcakes, 1 cupcake, or none of them (0 cupcakes).
For the violet frosting, she can take all 3 cupcakes, 2 cupcakes, 1 cupcake, or none of them (0 cupcakes).

Now let's calculate the total number of combinations:

For the red frosting, there are 2 options (either take all 4 cupcakes or none).
For the orange frosting, there are 2 options (either take both cupcakes or none).
For the yellow frosting, there are 2 options (either take both cupcakes or none).
For the green frosting, there are 2 options (either take both cupcakes or none).
For the blue frosting, there are 4 options (take all 3, 2, 1, or none).
For the violet frosting, there are 4 options (take all 3, 2, 1, or none).

To find the total number of combinations, we need to multiply the number of options for each color.

Total combinations = 2 (red) * 2 (orange) * 2 (yellow) * 2 (green) * 4 (blue) * 4 (violet)
= 256

Therefore, Julie could take 256 different combinations of cupcakes to her party.