space shuttle astronauts each consume an average of 3000 calories per day. one meal normally consists of a main dish, a vegetable dish, and 2 different desserts. the astronauts can choose from 11 man dishes, 8 vegetable dishes, and 12 desserts. how many different meals are possible?
To find the number of different meals possible, we need to multiply the number of choices for each component of the meal.
First, let's calculate the number of choices for the main dish. There are 11 main dishes available, so astronauts have 11 choices.
Next, let's calculate the number of choices for the vegetable dish. There are 8 vegetable dishes available, so astronauts have 8 choices for this component.
Finally, let's calculate the number of choices for the desserts. There are 12 different desserts available, and the astronauts can choose 2 desserts for each meal. We need to use combinations to calculate the number of different ways to choose 2 desserts from 12. This can be calculated using the formula: nCr = n! / (r! * (n - r)!)
Substituting the values, we get:
12C2 = 12! / (2! * (12 - 2)!)
= 12! / (2! * 10!)
= (12 * 11 * 10!) / (2! * 10!)
= (12 * 11) / (2 * 1)
= 132 / 2
= 66
So, there are 66 different choices for the desserts.
To get the total number of different meals, we need to multiply the number of choices for each component:
Total number of meals = Number of main dishes * Number of vegetable dishes * Number of dessert combinations
= 11 * 8 * 66
= 5,808
Therefore, there are 5,808 different meals possible for the astronauts.