Counting Combinations and Permutations - Example

A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses.

A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.

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

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

each other?

c) 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?

a) There is no order of precedence.

10 sopranos can be chosen from 13 indifferent ways.

9 sopranos can be chosen from 12 indifferent ways.

4 sopranos can be chosen from 6 indifferent ways.

4 sopranos can be chosen from 7 indifferent ways.

Hence the group can be chosen inways.

b) The 6 tallest sopranos can be arranged inways and the 4 shorter ones can be arranged in ways. Also the taller sopranos can occupy the first 1 to 6, 2 to 7, 3 to 8, 4 to 9 or 5 to 10 which introduces an additional factor of 5 so there areways.

c) If three of the tenors refuse to stand next to any of the basses then either

The remaining tenor stands next to the bases.

The ramining tenors can be arranged in 3! ways and the tenors in 4! ways. Also either the tenors can be grouped first or the basses which introduces an extra factor of 2, which makes 3!*4!*2 ways in which the basses and tenors can be arranged if they are grouped together. In addition the other 19 (10 sopranos and 9 altos) in the group can be arranged in 19! different ways, and the basses/tenors can occup positions 1 to 8, 2 to 9, …, 20 to 27 etc which introduces another factor of 20, so there are 3!*4!*2*19!*20 possible arrangements.

Or the tenors don't stand next to the basses.

Now the tenors may be arranged in 4! different ways, the basses in 4! different ways and the other 19 members of the group in 19! different ways.

If the basses occupy positions 1 to 4, the tenors may occupy positions 6 to 9, 7 to 10,...,24 to 27 which introduces a factor 19.

If the basses occupy positions 2 to 5, the tenors may occupy positions 7 to 10, 8 to 11,...,24 to 27 which introduces a factor 18.

If the basses occupy positions 3 to 6, the tenors may occupy positions 8 to 11, 9 to 12,...,24 to 27 which introduces a factor 17.

If the basses occupy positions 4 to 7, the tenors may occupy positions 9 to 12, 10 to 13,...,24 to 27 which introduces a factor 16.

We can go on in this manner obtaining an extra factor 19+18+17+16+15+14+13+12+11=10+9+8+7+6+5+4+3+2+1=190

We double this because we may interchange the tenors and basses.

Hence there are 4!*4!*19!*190 different ways of arranging the group if basses and tenors dont stand next to each other.

There are 3!*4!*2*19!*20+ 4!*4!*19!*190 ways altogether.