There is 12 people in a group .When two people are selected at random ,difference between probability of either both are male ,or both are female is `(1)/(6)` .Find difference between male and female in the group (male are more than female in the group)?

Options

A. 3
B. 2
C. 1
D. 4

Answer

To solve the problem step by step, we need to find the difference between the number of males and females in a group of 12 people, given that the difference between the probabilities of selecting two males and two females is \( \frac{1}{6} \).

### Step 1: Define Variables
Let:
- \( M \) = number of males
- \( F \) = number of females

From the problem, we know:
\[ M + F = 12 \]

### Step 2: Write the Probability Expressions
The probability of selecting 2 males is given by:
\[ P(\text{both male}) = \frac{M}{12} \times \frac{M-1}{11} \]
The probability of selecting 2 females is given by:
\[ P(\text{both female}) = \frac{F}{12} \times \frac{F-1}{11} \]

### Step 3: Set Up the Equation
According to the problem, the difference between these probabilities is \( \frac{1}{6} \):
\[ P(\text{both male}) - P(\text{both female}) = \frac{1}{6} \]
Substituting the expressions we derived:
\[ \frac{M(M-1)}{12 \times 11} - \frac{F(F-1)}{12 \times 11} = \frac{1}{6} \]

### Step 4: Simplify the Equation
Multiply through by \( 12 \times 11 \) to eliminate the denominators:
\[ M(M-1) - F(F-1) = \frac{1}{6} \times 12 \times 11 \]
Calculating the right side:
\[ \frac{1}{6} \times 12 \times 11 = 22 \]
So, we have:
\[ M(M-1) - F(F-1) = 22 \]

### Step 5: Substitute \( F \)
Since \( F = 12 - M \), we can substitute this into the equation:
\[ M(M-1) - (12-M)(11-M) = 22 \]

### Step 6: Expand and Simplify
Expanding \( (12-M)(11-M) \):
\[ (12-M)(11-M) = 132 - 23M + M^2 \]
Substituting this back into the equation:
\[ M(M-1) - (132 - 23M + M^2) = 22 \]
This simplifies to:
\[ M^2 - M - 132 + 23M - M^2 = 22 \]
Combining like terms:
\[ 22M - 132 = 22 \]
Adding 132 to both sides:
\[ 22M = 154 \]
Dividing by 22:
\[ M = 7 \]

### Step 7: Find \( F \)
Now substituting \( M \) back to find \( F \):
\[ F = 12 - M = 12 - 7 = 5 \]

### Step 8: Calculate the Difference
The difference between the number of males and females is:
\[ M - F = 7 - 5 = 2 \]

### Final Answer
The difference between the number of males and females in the group is **2**.

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