Assume that for every person the probability that he has exactly one child, exactly 2 children and exactly 3 children are `(1)/(4), (1)/(2) and (1)/(4)` respectively . The probability that a person will have 4 grand children can be expressed as `(p)/(q)` where p and q are relatively prime positive integers. Find the value of `5p-q`.

Answer

To solve the problem, we need to calculate the probability that a person will have exactly 4 grandchildren based on the given probabilities of having 1, 2, or 3 children.

### Step-by-Step Solution:

1. **Identify the probabilities**:
- Probability of having 1 child, \( P(1) = \frac{1}{4} \)
- Probability of having 2 children, \( P(2) = \frac{1}{2} \)
- Probability of having 3 children, \( P(3) = \frac{1}{4} \)

2. **Determine cases for 4 grandchildren**:
- A person cannot have 4 grandchildren if they have only 1 child or 3 children.
- Therefore, we only consider the case where the person has 2 children.

3. **Calculate the probability of having 4 grandchildren with 2 children**:
- If a person has 2 children, each child can have a certain number of children themselves.
- The combinations that lead to 4 grandchildren are:
- One child has 1 grandchild and the other has 3 grandchildren.
- Both children have 2 grandchildren each.

4. **Calculate the probabilities for these combinations**:
- **Case 1**: One child has 1 grandchild and the other has 3 grandchildren:
- Probability of first child having 1 child: \( P(1) = \frac{1}{4} \)
- Probability of second child having 3 children: \( P(3) = \frac{1}{4} \)
- Combined probability for this case:
\[ P(1) \cdot P(3) = \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16} \]
- Since either child can have 1 or 3 grandchildren, we multiply by 2 (for the two arrangements):
\[ 2 \cdot \frac{1}{16} = \frac{1}{8} \]

- **Case 2**: Both children have 2 grandchildren each:
- Probability of each child having 2 children: \( P(2) = \frac{1}{2} \)
- Combined probability for this case:
\[ P(2) \cdot P(2) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

5. **Total probability of having 4 grandchildren**:
- Combine the probabilities from both cases:
\[ P(4 \text{ grandchildren}) = \frac{1}{8} + \frac{1}{4} \]
- Convert \( \frac{1}{4} \) to eighths:
\[ \frac{1}{4} = \frac{2}{8} \]
- Thus,
\[ P(4 \text{ grandchildren}) = \frac{1}{8} + \frac{2}{8} = \frac{3}{8} \]

6. **Express the probability in the form \( \frac{p}{q} \)**:
- Here, \( p = 3 \) and \( q = 8 \).

7. **Find \( 5p - q \)**:
- Calculate \( 5p - q \):
\[ 5p - q = 5 \cdot 3 - 8 = 15 - 8 = 7 \]

### Final Answer:
The value of \( 5p - q \) is \( \boxed{7} \).