June 2022 p53 q7
2810
A group of 15 friends visit an adventure park. The group consists of four families.
- Mr and Mrs Kenny and their four children
- Mr and Mrs Lizo and their three children
- Mrs Martin and her child
- Mr and Mrs Nantes
The group enter the park by walking through a gate one at a time.
In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?
Solution
Since Mr Lizo goes first, we consider the remaining families as groups:
- Lizo family: 4 members (excluding Mr Lizo)
- Kenny family: 6 members
- Martin family: 2 members
- Nantes family: 2 members
First, calculate the number of ways to arrange the families:
\(3! = 6 ext{ ways (for Kenny, Martin, Nantes)}\)
Next, calculate the arrangements within each family:
\(4! = 24 ext{ ways (Lizo family)}\)
\(6! = 720 ext{ ways (Kenny family)}\)
\(2! = 2 ext{ ways (Martin family)}\)
\(2! = 2 ext{ ways (Nantes family)}\)
Multiply these together to find the total number of arrangements:
\(4! imes 6! imes 2! imes 2! imes 3! = 24 imes 720 imes 2 imes 2 imes 6 = 414,720\)
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