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  • A coffee shop has 20 different flavors of tea, including 6 black, 8 green, and 6 herbal teas. In how many ways can a customer select a set of 10 teas, where the set consists of exactly 3 black teas

A coffee shop has 20 different flavors of tea, including 6 black, 8 green, and 6 herbal teas. In how many ways can a customer select a set of 10 teas, where the set consists of exactly 3 black teas, 4 green teas, and 3 herbal teas, and no two teas of the same type can be adjacent in the selection?45,360

 

Option: 1

22,600


Option: 2

33,600


Option: 3

66,400


Option: 4

45,600


Answers (1)

best_answer

To solve this problem, we can break it down into multiple steps.

Step 1: Selecting the black teas

Since we need exactly 3 black teas in the set, we can select them in C(6, 3) = 20 ways.

Step 2: Selecting the green teas

Now, we have 10 - 3 = 7 teas remaining to be selected. Since we need exactly 4 green teas, we can select them in C(8, 4) = 70 ways.

Step 3: Selecting the herbal teas

We now have 7 - 4 = 3 teas remaining to be selected. Since we need exactly 3 herbal teas, we can select them in C(6, 3) = 20 ways.

Step 4: Arranging the selected teas

Now that we have selected the required number of teas from each category, we need to arrange them in a way that no two teas of the same type are adjacent.

Firstly, we can arrange the black teas in (3-1) !=2 ! ways to ensure they are not adjacent.
Next, we can arrange the green teas in (4-1) !=3 !  ways to ensure they are not adjacent.
Lastly, we can arrange the herbal teas in (3-1) !=2 ! ways to ensure they are not adjacent.
Step 5: Combining the steps
To get the total number of ways, we multiply the results from all the steps together:

\text { Total ways }=20 \times 70 \times 20 \times 2 ! \times 3 ! \times 2 !
=20 \times 70 \times 20 \times 2 \times 1 \times 3 \times 2 \times 1 \times 2 \times 1
=33,600

Therefore, a customer can select a set of 10 teas, consisting of exactly 3 black teas, 4 green te 3 herbal teas, where no two teas of the same type are adjacent, in the calculated total ways.

 

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