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June 2010 p63 q4
2801
Three identical cans of cola, 2 identical cans of green tea, and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if:
the first and last cans in the row are the same type of drink,
the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.
Solution
(i) Consider the cases where the first and last cans are the same type:
If they are cola, the arrangement is \\frac{5!}{2!2!} = 30\ ways.
If they are green tea, the arrangement is \\frac{5!}{3!2!} = 10\ ways.
If they are orange juice, the arrangement is \\frac{5!}{3!2!} = 10\ ways.
\(Total arrangements = 30 + 10 + 10 = 50\ ways.\)
\((ii) Treat the 3 cans of cola as a single block. The arrangement without restriction is \\frac{5!}{2!2!} = 30\ ways.\)
\(Now consider the case where both colas and green teas are together: \\frac{4!}{2!} = 12\ ways.\)
\(Subtract the restricted case from the total: 30 - 12 = 18\ ways.\)