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June 2013 p62 q6
2856
A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.
How many different selections of 12 trees can be made if there must be at least 2 of each type of tree?
Solution
We need to select 12 trees with at least 2 of each type: hibiscus (H), jacaranda (J), and oleander (O).
First, allocate 2 trees of each type: 2H, 2J, 2O. This uses up 6 trees, leaving 6 more to select.
Consider the following combinations for the remaining 6 trees:
1H, 8J, 2O: Choose 1 more hibiscus, 8 jacarandas, and 2 oleanders. The number of ways is given by:
\(\binom{4}{2} \times \binom{9}{8} \times \binom{2}{2} = 54\).
3H, 7J, 2O: Choose 3 hibiscus, 7 jacarandas, and 2 oleanders. The number of ways is given by:
\(\binom{4}{3} \times \binom{9}{7} \times \binom{2}{2} = 144\).
4H, 6J, 2O: Choose 4 hibiscus, 6 jacarandas, and 2 oleanders. The number of ways is given by:
\(\binom{4}{4} \times \binom{9}{6} \times \binom{2}{2} = 84\).
Summing these possibilities gives the total number of ways:
\(54 + 144 + 84 = 282\).
