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A dinner menu is to be designed out of 5 different starters, 6 identical main courses and 4 distinct desserts. In how many ways menu be designed such that there is atleast one of each of the starters, main courses and desserts?
- a)31 x 6 x 15
- b)32 x 6 x 16
- c)31 x 7 x 15
- d)5 x 6 x 4
Correct answer is option 'A'. Can you explain this answer?
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Problem Analysis:
To design the dinner menu, we need to select at least one starter, one main course, and one dessert. Let's analyze the different options we have for each category:
- Starters: There are 5 different options, and we need to select at least one.
- Main courses: There are 6 identical options, and we need to select at least one.
- Desserts: There are 4 distinct options, and we need to select at least one.
Calculating the Number of Ways:
To calculate the number of ways the menu can be designed, we can use the concept of combinations. We can select the starters, main courses, and desserts separately, and then multiply the number of options for each category.
1. Selecting starters:
- We have 5 different options, and we need to select at least one.
- The number of ways to select at least one starter is given by 2^5 - 1, which is equal to 31.
- The 2^5 term represents all possible combinations of starters (including selecting none), and subtracting 1 accounts for the case where no starters are selected.
2. Selecting main courses:
- We have 6 identical main courses, and we need to select at least one.
- The number of ways to select at least one main course is simply 6.
3. Selecting desserts:
- We have 4 distinct options, and we need to select at least one.
- The number of ways to select at least one dessert is given by 2^4 - 1, which is equal to 15.
- The 2^4 term represents all possible combinations of desserts (including selecting none), and subtracting 1 accounts for the case where no desserts are selected.
Calculating the Total Number of Ways:
To calculate the total number of ways the menu can be designed, we multiply the number of options for each category:
Total number of ways = Number of ways to select starters * Number of ways to select main courses * Number of ways to select desserts
= 31 * 6 * 15
= 31 * 90
= 2790
Therefore, the correct answer is option A) 31 x 6 x 15.
To design the dinner menu, we need to select at least one starter, one main course, and one dessert. Let's analyze the different options we have for each category:
- Starters: There are 5 different options, and we need to select at least one.
- Main courses: There are 6 identical options, and we need to select at least one.
- Desserts: There are 4 distinct options, and we need to select at least one.
Calculating the Number of Ways:
To calculate the number of ways the menu can be designed, we can use the concept of combinations. We can select the starters, main courses, and desserts separately, and then multiply the number of options for each category.
1. Selecting starters:
- We have 5 different options, and we need to select at least one.
- The number of ways to select at least one starter is given by 2^5 - 1, which is equal to 31.
- The 2^5 term represents all possible combinations of starters (including selecting none), and subtracting 1 accounts for the case where no starters are selected.
2. Selecting main courses:
- We have 6 identical main courses, and we need to select at least one.
- The number of ways to select at least one main course is simply 6.
3. Selecting desserts:
- We have 4 distinct options, and we need to select at least one.
- The number of ways to select at least one dessert is given by 2^4 - 1, which is equal to 15.
- The 2^4 term represents all possible combinations of desserts (including selecting none), and subtracting 1 accounts for the case where no desserts are selected.
Calculating the Total Number of Ways:
To calculate the total number of ways the menu can be designed, we multiply the number of options for each category:
Total number of ways = Number of ways to select starters * Number of ways to select main courses * Number of ways to select desserts
= 31 * 6 * 15
= 31 * 90
= 2790
Therefore, the correct answer is option A) 31 x 6 x 15.
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A dinner menu is to be designed out of 5 different starters, 6 identical main courses and 4 distinct desserts. In how many ways menu be designed such that there is atleast one of each of the starters, main courses and desserts?a)31 x 6 x 15b)32 x 6 x 16c)31 x 7 x 15d)5 x 6 x 4Correct answer is option 'A'. Can you explain this answer?
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