The number of employees in a nationalised bank in a small town is 10, out of which 4 are female and the rest are males. A committee of 5 is to be formed. If m be the number of ways to form such a committee in which there is atleast one female employee and n be the no. of ways to form such a committee which includes at least two male employees, then find the ratio m : n.

Options

A. `3 : 2`
B. `5 : 2`
C. `1 : 1`
D. `8 : 9`

Answer

To solve the problem, we need to calculate the values of \( m \) and \( n \) separately and then find the ratio \( m : n \).

### Step 1: Calculate \( m \)
\( m \) is the number of ways to form a committee of 5 employees with at least one female employee.

1. **Total Employees**: There are 10 employees in total (4 females and 6 males).
2. **Case Analysis**: We can use complementary counting. First, we calculate the total ways to form a committee of 5 employees without any restriction and then subtract the cases where there are no female employees (i.e., all members are male).

- **Total ways to choose 5 from 10**:
\[ \text{Total ways} = \binom{10}{5} \]

- **Ways to choose 5 males from 6** (no females):
\[ \text{Ways with no females} = \binom{6}{5} \]

- **Calculate \( m \)**:
\[ m = \binom{10}{5} - \binom{6}{5} \]

### Step 2: Calculate \( n \)
\( n \) is the number of ways to form a committee of 5 employees with at least 2 male employees.

1. **Case Analysis**: We can use complementary counting again. We can calculate the total ways to form a committee of 5 employees and subtract the cases where there are fewer than 2 males (i.e., 0 or 1 male).

- **Ways with 0 males (all females)**:
\[ \text{Ways with 0 males} = \binom{4}{5} = 0 \quad (\text{not possible}) \]

- **Ways with 1 male**:
- Choose 1 male from 6:
\[ \binom{6}{1} \]
- Choose 4 females from 4:
\[ \binom{4}{4} \]
- Total for this case:
\[ \text{Ways with 1 male} = \binom{6}{1} \cdot \binom{4}{4} \]

- **Calculate \( n \)**:
\[ n = \binom{10}{5} - \left( \binom{6}{1} \cdot \binom{4}{4} \right) \]

### Step 3: Calculate the values
1. **Calculate \( m \)**:
\[ m = \binom{10}{5} - \binom{6}{5} = 252 - 6 = 246 \]

2. **Calculate \( n \)**:
\[ n = \binom{10}{5} - \left( \binom{6}{1} \cdot \binom{4}{4} \right) = 252 - (6 \cdot 1) = 252 - 6 = 246 \]

### Step 4: Find the ratio \( m : n \)
\[ \text{Ratio } m : n = 246 : 246 = 1 : 1 \]

### Final Answer
The ratio \( m : n \) is \( 1 : 1 \).

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