We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can the selections be made? 15 (b) 22 (c) 7 (d) None of these

Answer

To solve the problem of selecting 6 persons from 8 with the condition that if person A is chosen, then person B must also be chosen, we can break it down into two cases:

### Step 1: Define the total number of persons and selections
We have a total of 8 persons, and we need to select 6 persons.

### Step 2: Case 1 - A is chosen
If person A is chosen, then person B must also be chosen. This means we have already selected 2 persons (A and B), and we need to select 4 more persons from the remaining 6 persons (since A and B are already chosen).

**Calculation for Case 1:**
- Remaining persons to choose from = 8 - 2 = 6
- We need to choose 4 persons from these 6.
- The number of ways to choose 4 persons from 6 is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).

So, we calculate:
\[ C(6, 4) = \frac{6!}{4! \cdot (6-4)!} = \frac{6 \cdot 5}{2 \cdot 1} = 15 \]

### Step 3: Case 2 - A is not chosen
If person A is not chosen, we can select any 6 persons from the remaining 7 persons (B is included in this case).

**Calculation for Case 2:**
- Remaining persons to choose from = 8 - 1 = 7
- We need to choose 6 persons from these 7.

So, we calculate:
\[ C(7, 6) = \frac{7!}{6! \cdot (7-6)!} = \frac{7}{1} = 7 \]

### Step 4: Total number of ways
Now, we add the number of ways from both cases to get the total number of selections:
\[ \text{Total ways} = C(6, 4) + C(7, 6) = 15 + 7 = 22 \]

### Final Answer
Thus, the total number of ways to select 6 persons from 8, considering the given conditions, is **22**.

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