Statistics Help
Space shuttle astronauts each consume an average of 3000 calories per day. One meal normally consists of a main dish, a vegetable dish, and two different desserts. The astronaus can choose from 10 main dishes, 8 vegetable dishes, and 13 desserts. How many different meals are possible.
The back of the book says 6240, but I have no idea how they got that answer.
To find the number of different meals that are possible, we need to multiply the number of options for each component.
The number of options for the main dish is 10, the number of options for the vegetable dish is 8, and the number of options for desserts is 13.
To find the total number of meals, we multiply these numbers together:
10 x 8 x 13 = 1040
However, this calculation only accounts for one particular combination of a main dish, a vegetable dish, and two desserts. To find the total number of possible combinations, we need to consider all possible combinations of dishes.
To do this, we multiply the total number of possible dishes for each component. For the main dish, we have 10 options. For the vegetable dish, we have 8 options. And for the desserts, we have 13 options.
Since we have two desserts, we consider choosing two different desserts as distinct from choosing the same dessert twice.
Therefore, we take the product of the number of options for each component and multiply it by the number of ways to choose two desserts:
10 x 8 x (13 choose 2) = 10 x 8 x 78 = 6240
So, the total number of different meals that are possible is 6240.
To find the number of different meals that are possible, we need to determine the number of combinations of main dishes, vegetable dishes, and desserts.
Step 1: Determine the number of combinations for each component:
- There are 10 options for the main dish.
- There are 8 options for the vegetable dish.
- There are 13 options for the dessert.
Step 2: Multiply the number of combinations for each component together:
10 (main dishes) * 8 (vegetable dishes) * 13 (desserts) = 1,040.
However, this answer represents all possible combinations, including cases where the same main dish, vegetable dish, and dessert are selected multiple times in a meal.
Step 3: Exclude duplicate combinations:
Since each meal consists of one main dish, one vegetable dish, and two desserts, we need to divide the total number of combinations by the number of possible arrangements of desserts (which is 2).
1,040 (total combinations) / 2 (number of arrangements for desserts) = 520.
So, there are 520 different meals possible.
It seems that the answer in the book might be incorrect. The correct answer should be 520, not 6240.
The tricky part is the number of ways (combinations) of choosing 2 DIFFERENT desserts from 13. It does not matter in what order the two desserts are picked. That number of combinations is
13!/(11!*2!) = (13*12)/2 = 78.
We are assuming that they are required to have two different desserts. Zero or one dessert is "not an option", as they say.
Now multiply that 78 by the possible number of main dishes (8) and the number of vegetable dishes (10). You will get the answer.