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June 2009 p6 q4
2804
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.
(ii) In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
(iii) 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?
Solution
(ii) Consider the 6 tallest sopranos as a single block. This block can be arranged in a line with the remaining 4 sopranos. There are 5 blocks in total (1 block of 6 sopranos and 4 individual sopranos). The number of ways to arrange these 5 blocks is given by:
\(5!\)
Within the block of 6 sopranos, they can be arranged among themselves in:
\(6!\)
Therefore, the total number of arrangements is:
\(5! \times 6! = 86400\)
(iii) Consider the 4 tenors as a single block and the 4 basses as another block. These two blocks can be arranged in:
\(2!\)
Within the block of 4 tenors, they can be arranged among themselves in:
\(4!\)
Within the block of 4 basses, they can be arranged among themselves in: