Question

you are eating smarties. you have decided to limit yourself to merely 20 smarties. there are 6 colours in the box; red, orange, yellow, mauve, pink and brown. (a) assuming there are more than 20 smarties of each colour, in how many different colour combinations can you choose your smarties from the box. (b) assuming there are more than 20 smarties of each colour, in how many different colour combinations can you choose your smarties such that you have at least 3 red ones. (c) now suppose that there are only 10 of the red smarties in the box but at least 20 of the other colours. now how many different colour combination are there for you to choose your 20 smarties?

Answer 1

For (a) assuming there are more than 20 smarties of each color, there are 64 different color combinations. For (b) assuming there are more than 20 smarties of each color and at least 3 red ones, there are 128 different color combinations. For (c) assuming there are only 10 red smarties and at least 20 of the other colors, there are 312,500 different color combinations.

Step-by-step explanation:

(a) To find the number of different color combinations you can choose from, you need to consider that for each color (red, orange, yellow, mauve, pink, and brown) there are more than 20 smarties available in the box. So for each color, you have the choice to either include it in your selection or not. Therefore, the total number of different color combinations can be calculated as 2^6 = 64, where 6 represents the number of colors.

(b) If you want to select 20 smarties with at least 3 red ones, you can consider the remaining 17 smarties. Now, you have the same choices as before for the remaining colors (orange, yellow, mauve, pink, and brown), but you must include at least 3 red ones. This means you have 4 choices for the remaining red smarties (3, 4, 5, or 6) and 2 choices for each of the other colors (include or exclude). Therefore, the total number of different color combinations can be calculated as 4*2^5 = 128.

(c) In this case, there are only 10 red smarties available. Therefore, you can choose all 10 of them, and then select the remaining 10 smarties from the other colors (orange, yellow, mauve, pink, and brown). This means you have 5 choices for each of the other colors (include or exclude). Therefore, the total number of different color combinations can be calculated as 10*5^5 = 312,500.