Assume that you have 6 different types of vegetables and 9 different types of fruit. A vegetable salad consists of a mixture of any 3 types of vegetables, and a fruit salad consists of a mixture of 3 kinds of fruit.
(1) In how many different ways can you prepare a vegetable salad and a fruit salad?
(2) In how many different ways can you prepare a vegetable salad or a fruit salad but not both?
(3) How many ways can you choose two different kinds of salad?
To solve these problems, we can use the concepts of combinations and counting principles. Here are the step-by-step solutions to each question:
(1) In how many different ways can you prepare a vegetable salad and a fruit salad?
To prepare the vegetable salad, you need to choose any 3 types of vegetables out of the 6 available. This can be calculated using a combination formula:
Number of ways to choose 3 vegetables out of 6 = 6C3 = 6! / (3! * (6-3)!) = 20.
Similarly, to prepare the fruit salad, you need to choose any 3 types of fruit out of the 9 available:
Number of ways to choose 3 fruits out of 9 = 9C3 = 9! / (3! * (9-3)!) = 84.
To find the number of different ways to prepare both salads, you need to multiply the number of ways for each salad:
Total number of different ways = Number of ways for vegetable salad * Number of ways for fruit salad = 20 * 84 = 1680.
Therefore, you can prepare a vegetable salad and a fruit salad in 1680 different ways.
(2) In how many different ways can you prepare a vegetable salad or a fruit salad but not both?
To find the number of ways to prepare a vegetable salad or a fruit salad but not both, you need to subtract the number of ways to prepare both salads from the total number of ways to prepare each salad individually.
Total number of ways to prepare a vegetable salad = Number of ways for vegetable salad = 20.
Total number of ways to prepare a fruit salad = Number of ways for fruit salad = 84.
Number of ways to prepare both salads = Total number of different ways = 1680.
Number of ways to prepare either a vegetable salad or a fruit salad but not both = Total number of ways for vegetable salad + Total number of ways for fruit salad - Number of ways to prepare both salads.
= 20 + 84 - 1680 = -1576.
However, a negative number of ways does not make sense in this context. Therefore, there are no different ways to prepare either a vegetable salad or a fruit salad but not both.
(3) How many ways can you choose two different kinds of salad?
To choose two different kinds of salad, you can either have a vegetable salad and a fruit salad, or you can have only a vegetable salad or only a fruit salad.
Number of ways to choose two different kinds of salad = Number of ways to prepare both salads + Number of ways to prepare either a vegetable salad or a fruit salad but not both.
= 1680 + 0 = 1680.
Therefore, there are 1680 different ways to choose two different kinds of salad.
To find the answers to these questions, we can use the concept of combinations.
1) In how many different ways can you prepare a vegetable salad and a fruit salad?
To find the number of ways to prepare a vegetable salad, we need to choose 3 types of vegetables from the 6 different types. This can be done using the combination formula:
C(6, 3) = 6! / (3! * (6 - 3)!) = 20
Similarly, for the fruit salad, we need to choose 3 types of fruit from the 9 different types:
C(9, 3) = 9! / (3! * (9 - 3)!) = 84
To prepare both salads, we need to multiply the number of ways of preparing each salad:
20 * 84 = 1,680
So, there are 1,680 different ways to prepare both a vegetable salad and a fruit salad.
2) In how many different ways can you prepare a vegetable salad or a fruit salad but not both?
To find the number of ways to prepare either a vegetable salad or a fruit salad, we need to consider the cases where the other salad is not prepared.
For the vegetable salad, we have already calculated that there are 20 different ways of preparing it. So, if we don't prepare the vegetable salad, we can choose any combination of fruits, which is:
C(9, 3) = 84
Similarly, if we don't prepare the fruit salad, we can choose any combination of vegetables, which is:
C(6, 3) = 20
Since we want to find the ways to prepare either one but not both, we need to subtract the number of ways of preparing both salads from the total:
(20 + 84) - 1,680 = 76
So, there are 76 different ways to prepare a vegetable salad or a fruit salad, but not both.
3) How many ways can you choose two different kinds of salad?
To find the number of ways to choose two different kinds of salad, we can calculate the combinations of selecting a vegetable salad and a fruit salad.
From the calculations above, we know that there are 1,680 ways to prepare both salads. So, the number of ways to choose two different kinds of salad is the same:
1,680 ways
Therefore, there are 1,680 ways to choose two different kinds of salad.
(1) first
Veg
combinations of 6 things taken 3 at a time
= 6!/[3!(6-3)! ] = 6*5*4/(3*2) = 20
Fruit
= 9!/[ 3! (9-3)! ] = 9*8*7*6*5*4/(6*5*4*3*2)
= 9*8*7/6
= 12*7 = 84
(2) 20+84 = 104
(3) 20 kinds of veg salad and 84 of fruit salad
for veg 1, 84 different fruits
for veg 2, 84 different fruits
etc
so 20* 84 = 1680