a choir consist of 13 sopranos, 12 altos, 6 tenors and 7 bases. A group consisti

Question

a choir consist of 13 sopranos, 12 altos, 6 tenors and 7 bases. A group consisting 10 sopranos, 9 altos, 4 tenors and 4 bases is to be chosen from the choir 1) how many different ways can the group be chosen? 2) in how many different ways can the 10 sopranos be arranged in a line if the 6 shortest stand next to each other? 3) the 4 tenors and 4 bases in the group stand in a line with all the tenors next to each other. how many possible arrangements are there if three of the tenors refuse to stand next to any of the bases

Explanation / Answer

If you don't already know the basics of permutations and combinations, and the different formulas involved, I highly recommend the source linked below. It is an excellent and thorough explanation and includes one case for which good explanations are scarce.

1) In how many different ways can the group be chosen?

We have a formula for combinations, which we can apply to the choice of singers in each section. Since each combination for any section can be combined with any set of combinations from the other sections, we have to multiply them.

Sopranos: 13C10 = 13! / (10! 3!)
= 13 * 12 * 11 / (3 * 2 * 1) = 13 * 2 * 11 = 286

Altos: 12C9 = 12! / (9! 3!)
= 12 * 11 * 10 / (3 * 2 * 1) = 2 * 11 * 10 = 220

Tenors: 6C4 = 15

Basses: 7C4 = 35

Whole group:
286 * 220 * 15 * 35 = 33,033,000


2) In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?

Now we want two permutations, one for the 6 tallest sopranos and one for the other 4. But there's a catch here: we aren't told that the 4 shortest ones are all together, just the 6 tallest ones. So the group of 6 tallest ones can be inserted anywhere, as one unit, in the line of the shortest ones, including either end.

Permutations of 6 tallest: 6P6 = 6! = 720
Permutations of 4 shortest: 4P4 = 4! = 24
Places the group of tallest can be: 5

Total ways: 720 * 24 * 5 = 86,400


3) The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?

Obviously, the one tenor who won't get thrown off by standing next to the basses has to be the one next to them. The basses are either to his left or his right, with the other three tenors on his other side.

Permutations of 4 basses: 4! = 24
Permutations of 3 movable tenors: 3! = 6
Basses left or right side? 2 choices

Total arrangements of tenors and basses: 24 * 6 * 2 = 288

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