A choir consists of 13 sapranos,12 altos,6 tenors, and 7 basses.A group consisting of 10 sapranos, 9 altos, 4 tenors and 7 basses is to be chosen from the choir.
Q.The 4 tenors and 4 basses stand in a single line with all the tenors next to each other and all the basses next to each other.How many possible arrangement are there if 3 of the tenors refuse to stand next to any of the basses?
there are 13C10 ways to pick the 10 sopranos
and so on for the other voices.
Now just multiply all those combinations together.
There is only 1 tenor who will stand next to the basses.
There are 3! ways to arrange the oher tenors, and 4! ways to arrange the basses.
Then there are two ways to arrange the sections: Tenors-basses and basses-tenors.
So, 2*3!*4! ways in all.
Well, well, well. Looks like we have some choir drama going on here! 🎵 Let me do some quick calculations for you.
We need to choose 4 tenors and 7 basses for this arrangement. However, the 3 tenors are being a bit picky and don't want to stand next to any of the basses.
To solve this, we can first consider the 3 "rebellious" tenors as a single entity. So now we have 1 group of 3 tenors, 1 lonely tenor, and the 7 basses.
Let's arrange them in this order: (3 tenors) - (1 tenor) - (7 basses).
The 3 tenors can be arranged amongst themselves in 3! (that's 3 factorial) ways. The lonely tenor has only one place to go, right in between the group of 3 and the 7 basses.
So that leaves us with 10 sapranos, 9 altos, and 6 remaining tenors to arrange. They can be arranged in (10 + 9 + 6)! ways.
Now we just multiply both numbers to get the total number of arrangements: 3! * (10 + 9 + 6)!
And if you calculate that, you'll have your answer!
To solve this question, we can consider the following steps:
Step 1: Determine the number of arrangements for the tenors and basses individually:
- Since there are 6 tenors and 4 of them need to stand together, we can treat them as a single entity, resulting in 3 tenors.
- Similarly, there are 7 basses, and all of them need to stand together.
Step 2: Calculate the possible arrangements for the remaining choir members:
- The number of possible arrangements for the remaining choir members (sopranos and altos) can be calculated using combinatorics.
- From the given information, we know that there are 13 sopranos and 9 sopranos are to be chosen. Therefore, the number of ways to choose 9 sopranos from 13 can be calculated using the combination formula: C(13,9) = 13! / (9!(13-9)!) = 715.
- Similarly, there are 12 altos, and 9 altos are to be chosen. Therefore, the number of ways to choose 9 altos from 12 can be calculated using the combination formula: C(12,9) = 12! / (9!(12-9)!) = 220.
Step 3: Calculate the total number of possible arrangements:
- Now, we multiply the calculated number of arrangements for each group:
- Number of arrangements for the tenors and basses: 1 (since they are already standing together).
- Number of arrangements for the sopranos: 715.
- Number of arrangements for the altos: 220.
- Therefore, the total number of possible arrangements is: 1 x 715 x 220 = 157,700.
Answer: There are 157,700 possible arrangements if 3 of the tenors refuse to stand next to any of the basses.
To answer this question, we need to consider the arrangements while taking into account the restriction that three tenors refuse to stand next to any of the basses.
First, let's consider the arrangements of the tenors and basses together. We have a total of 7 basses, so the number of ways to arrange them is 7!. Similarly, there are 4 tenors, so the number of ways to arrange them is 4!.
Since we want all the tenors to be together and all the basses to be together, we can consider the arrangement of the grouped tenors and the grouped basses as a single block. So, we have three groups to arrange: 1 block of tenors, 1 block of basses, and the remaining 13 sapranos and 12 altos.
The number of ways to arrange the three groups can be found by treating them as indistinguishable objects and applying factorial division. Thus, we divide by the factorial of the number of arrangements within each group:
Number of arrangements = (7! * 4!) / (3! * 1! * 12! * 13!)
Simplifying this expression, we have:
Number of arrangements = (7! * 4!) / (3! * 12! * 13!)
Now, we can calculate this value to find the number of possible arrangements.