Q64 / 180 · COMEDK 2025 (25 May Shift) · Mathematics
From a group of 8 people 6 are to be chosen such that if a person $A$ is chosen person $B$ must also be chosen. The number of selections that can be made is
- A8
- B15
- C7
- Correct answerD22
Solution
The total number of people is 8. We need to choose 6 people from these 8. Let the set of people be $\{A, B, C, D, E, F, G, H\}$. The condition is that if $A$ is chosen, $B$ must be chosen. This implies that the selection cannot contain $A$ without $B$.
We analyze the possible cases based on the inclusion of $A$ and $B$:
Case 1: $A$ is chosen. Since $A$ is chosen, $B$ must also be chosen. We have already selected 2 people ($A$ and $B$). We need to select $6 - 2 = 4$ more people from the remaining $8 - 2 = 6$ people. The number of ways is $^{6}C_{4} = ^{6}C_{2} = \dfrac{6 \times 5}{2 \times 1} = 15$.
Case 2: $A$ is not chosen. We need to select 6 people from the remaining $8 - 1 = 7$ people (excluding $A$). The number of ways is $^{7}C_{6} = ^{7}C_{1} = 7$.
The total number of selections is the sum of the ways from these two mutually exclusive cases: $15 + 7 = 22$.
Answer: 22
We analyze the possible cases based on the inclusion of $A$ and $B$:
Case 1: $A$ is chosen. Since $A$ is chosen, $B$ must also be chosen. We have already selected 2 people ($A$ and $B$). We need to select $6 - 2 = 4$ more people from the remaining $8 - 2 = 6$ people. The number of ways is $^{6}C_{4} = ^{6}C_{2} = \dfrac{6 \times 5}{2 \times 1} = 15$.
Case 2: $A$ is not chosen. We need to select 6 people from the remaining $8 - 1 = 7$ people (excluding $A$). The number of ways is $^{7}C_{6} = ^{7}C_{1} = 7$.
The total number of selections is the sum of the ways from these two mutually exclusive cases: $15 + 7 = 22$.
Answer: 22
Hence, the correct answer is option: 'D'.