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June 2015 p62 q6
2847
Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together.
Solution
First, calculate the total number of ways to select 9 people from 14 without any restrictions:
\(\binom{14}{9} = 2002\).
Next, calculate the number of ways where the two particular people (say T and M) are both included in the group. If T and M are both in the group, we need to select 7 more people from the remaining 12:
\(\binom{12}{7} = 792\).
Subtract the restricted cases from the total:
\(2002 - 792 = 1210\).
Alternatively, consider the cases where neither or only one of T or M is in the group:
Neither T nor M in the group: \(\binom{12}{9} = 220\).