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
To determine the different combinations of cupcakes that Julie could take to her party, we need to consider the number of cupcakes for each frosting color.
Based on the given information, here are the available cupcakes for each color:
- Red: 4 cupcakes
- Orange: 2 cupcakes
- Yellow: 2 cupcakes
- Green: 2 cupcakes
- Blue: 3 cupcakes
- Violet: 3 cupcakes
Now let's analyze the possible combinations:
1. Julie can take all 4 red cupcakes or none.
2. Julie can take both orange cupcakes or none.
3. Julie can take both yellow cupcakes or none.
4. Julie can take both green cupcakes or none.
5. Julie can take all 3 blue cupcakes or none.
6. Julie can take all 3 violet cupcakes or none.
As Julie plans to take exactly 10 cupcakes to her party, we can list out the possible combinations using the available cupcakes:
- 4 red cupcakes + 2 orange cupcakes + 2 yellow cupcakes + 2 green cupcakes = 10 cupcakes
- 4 red cupcakes + 2 orange cupcakes + 2 yellow cupcakes + 3 blue cupcakes = 11 cupcakes
- 4 red cupcakes + 2 orange cupcakes + 2 yellow cupcakes + 3 violet cupcakes = 12 cupcakes
- 4 red cupcakes + 2 orange cupcakes + 3 blue cupcakes + 3 violet cupcakes = 12 cupcakes
- 4 red cupcakes + 2 yellow cupcakes + 2 green cupcakes + 3 blue cupcakes = 11 cupcakes
- 4 red cupcakes + 2 yellow cupcakes + 2 green cupcakes + 3 violet cupcakes = 12 cupcakes
- 4 red cupcakes + 2 green cupcakes + 3 blue cupcakes + 3 violet cupcakes = 12 cupcakes
- 4 red cupcakes + 3 blue cupcakes + 3 violet cupcakes = 10 cupcakes
Therefore, there are 8 different combinations of cupcakes that Julie could take to her party.
If Julie includes one of the colors that cover three cupcakes, she must also include the other color that covers three cupcakes. This is because she must make ten cupcakes total, and all of the other colors cover an even number of cupcakes, so there is no way to make ten with three and some combination of even numbers. Thus, if she includes blue and violet, she has four cupcakes left to choose. There are three ways in which she can choose four cupcakes if she chooses colors that cover two (green and orange, green and yellow, or orange and yellow). Alternately, she can choose a color that covers four (red). Finally, if she doesn't include any colors that cover three cupcakes, she must choose all of the other cupcakes in order to make ten. Thus, Julie has 5 different combinations of cupcakes.
4+3+3=10
4+2+2+2=10
2+2+3+3=10
So figure out the ways to make those combinations. I get
1+1+3=5 ways to do it (3C2=3)