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June 2023 p51 q2
2848
(a) Find the number of ways in which a committee of 6 people can be chosen from 6 men and 8 women if it must include 3 men and 3 women.
A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
(b) Find the number of ways in which this committee can be chosen if there are no restrictions on the numbers of men and women, but it must include no more than two of the brothers.
Solution
(a) To form a committee of 6 people with 3 men and 3 women from 6 men and 8 women, we use combinations:
The number of ways to choose 3 men from 6 is given by \(^6C_3\).
The number of ways to choose 3 women from 8 is given by \(^8C_3\).
Thus, the total number of ways is \(^6C_3 \times ^8C_3 = 20 \times 56 = 1120\).
(b) We need to form a committee of 6 people with no more than 2 brothers. We consider the following cases:
0 brothers: Choose 0 brothers from 3 and 6 others from the remaining 11 people (3 non-brothers + 8 women): \(^3C_0 \times ^{11}C_6 = 1 \times 462 = 462\).
1 brother: Choose 1 brother from 3 and 5 others from the remaining 11 people: \(^3C_1 \times ^{11}C_5 = 3 \times 462 = 1386\).
2 brothers: Choose 2 brothers from 3 and 4 others from the remaining 11 people: \(^3C_2 \times ^{11}C_4 = 3 \times 330 = 990\).
Adding these cases gives the total number of ways: \(462 + 1386 + 990 = 2838\).
