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June 2021 p51 q1
2818
A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue.
How many different selections of 5 marbles contain at least 4 marbles of the same colour?
Solution
To find the number of different selections of 5 marbles containing at least 4 marbles of the same color, we consider two cases:
4 red marbles and 1 blue marble:
The number of ways to choose 4 red marbles from 8 is given by \(\binom{8}{4}\), and the number of ways to choose 1 blue marble from 4 is \(\binom{4}{1}\).
Thus, the total number of ways for this case is \(\binom{8}{4} \times \binom{4}{1} = 70 \times 4 = 280\).
4 blue marbles and 1 red marble:
The number of ways to choose 4 blue marbles from 4 is \(\binom{4}{4}\), and the number of ways to choose 1 red marble from 8 is \(\binom{8}{1}\).
Thus, the total number of ways for this case is \(\binom{4}{4} \times \binom{8}{1} = 1 \times 8 = 8\).
5 red marbles:
The number of ways to choose 5 red marbles from 8 is \(\binom{8}{5}\).
Thus, the total number of ways for this case is \(\binom{8}{5} = 56\).
Adding all these cases together, the total number of selections is:
