How many groups of 6 persons can be formed from 8 men and 7 women ?

Options

A. 5000
B. 5005
C. 5050
D. None of these

Answer

To solve the problem of how many groups of 6 persons can be formed from 8 men and 7 women, we can follow these steps:

### Step 1: Determine the total number of people
First, we need to find the total number of people available for selection. We have:
- 8 men
- 7 women

Total number of people = 8 + 7 = 15

### Step 2: Use the combination formula
We need to select 6 persons from the total of 15. The number of ways to choose \( r \) objects from \( n \) objects is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
In our case, \( n = 15 \) and \( r = 6 \).

### Step 3: Plug in the values into the formula
Using the combination formula, we have:
\[ C(15, 6) = \frac{15!}{6!(15 - 6)!} = \frac{15!}{6! \cdot 9!} \]

### Step 4: Simplify the factorial expression
We can simplify \( \frac{15!}{6! \cdot 9!} \) as follows:
\[ C(15, 6) = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9!}{6! \cdot 9!} \]
The \( 9! \) in the numerator and denominator cancels out:
\[ C(15, 6) = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6!} \]

### Step 5: Calculate \( 6! \)
Now we calculate \( 6! \):
\[ 6! = 720 \]

### Step 6: Calculate the numerator
Now we need to calculate the numerator:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]
\[ 32760 \times 11 = 360360 \]
\[ 360360 \times 10 = 3603600 \]

### Step 7: Divide the numerator by \( 6! \)
Now we divide the numerator by \( 720 \):
\[ C(15, 6) = \frac{3603600}{720} = 5005 \]

### Final Answer
Thus, the number of groups of 6 persons that can be formed from 8 men and 7 women is **5005**.

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