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Nov 2012 p61 q7
2791
(a) In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together.
(b) Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.
How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account?
The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.
How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen?
Solution
(a) Consider the packets of chocolates as a single unit. This gives us 5 packets of toffees, 4 packets of fruit gums, and 1 unit of chocolates, totaling 10 units to arrange. The number of arrangements is given by:
\(\frac{10!}{5!4!} = 1260\)
(b)(i) Jessica chooses 4 packets out of 8, considering the order. This is a permutation problem:
\({}_8P_4 = 8 \times 7 \times 6 \times 5 = 1680\)
(b)(ii) If the packets of chocolate biscuits and custard creams are chosen, we need to choose 2 more packets from the remaining 6. The number of ways to choose 2 packets from 6 is:
\(\binom{6}{2} = 15\)
Considering the order of 4 packets, we multiply by \(4!\):