A class contains 4 boys and `g` girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being set every week. During, the picnic, the class teacher gives a doll to each girl in the group. If the total number of dolls distributed was 85, then value of `g` is 15 (b) 12 (c) 8 (d) 5

Options

A. 15
B. 12
C. 8
D. 5

Answer

To solve the problem step by step, we will analyze the given information and use combinations to find the value of `g` (the number of girls).

### Step 1: Understand the problem
We have 4 boys and `g` girls in a class. Every Sunday, a group of 5 students is selected for a picnic, and it must include at least 3 boys. The total number of dolls distributed to the girls in the selected groups is 85.

### Step 2: Identify the possible groups
Since we need at least 3 boys in the group of 5, the possible combinations of boys and girls can be:
1. 3 boys and 2 girls
2. 4 boys and 1 girl

### Step 3: Calculate the number of ways to choose the groups
1. **Case 1: 3 boys and 2 girls**
- The number of ways to choose 3 boys from 4:
\[ \binom{4}{3} = 4 \]
- The number of ways to choose 2 girls from `g`:
\[ \binom{g}{2} \]
- Each girl receives 1 doll, so the total number of dolls given in this case is:
\[ 2 \times \binom{4}{3} \times \binom{g}{2} = 2 \times 4 \times \binom{g}{2} = 8 \binom{g}{2} \]

2. **Case 2: 4 boys and 1 girl**
- The number of ways to choose 4 boys from 4:
\[ \binom{4}{4} = 1 \]
- The number of ways to choose 1 girl from `g`:
\[ \binom{g}{1} = g \]
- Each girl receives 1 doll, so the total number of dolls given in this case is:
\[ 1 \times \binom{4}{4} \times \binom{g}{1} = 1 \times 1 \times g = g \]

### Step 4: Set up the equation
The total number of dolls distributed is the sum from both cases:
\[ 8 \binom{g}{2} + g = 85 \]

### Step 5: Simplify the equation
Now, we can express \(\binom{g}{2}\):
\[ \binom{g}{2} = \frac{g(g-1)}{2} \]
Substituting this into the equation gives:
\[ 8 \left(\frac{g(g-1)}{2}\right) + g = 85 \]
This simplifies to:
\[ 4g(g-1) + g = 85 \]
\[ 4g^2 - 4g + g - 85 = 0 \]
\[ 4g^2 - 3g - 85 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \(g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 4\), \(b = -3\), and \(c = -85\):
\[ g = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-85)}}{2 \cdot 4} \]
\[ g = \frac{3 \pm \sqrt{9 + 1360}}{8} \]
\[ g = \frac{3 \pm \sqrt{1369}}{8} \]
\[ g = \frac{3 \pm 37}{8} \]
Calculating the two possible values:
1. \(g = \frac{40}{8} = 5\)
2. \(g = \frac{-34}{8}\) (not valid since `g` must be non-negative)

Thus, the only valid solution is:
\[ g = 5 \]

### Final Answer
The value of `g` is **5**.

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