In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

Options

A. 63
B. 90
C. 126
D. 45

Answer

To solve the problem of how many ways a group of 5 men and 2 women can be formed from a total of 7 men and 3 women, we can use the concept of combinations. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the total number of men and women
We have:
- Total men = 7
- Total women = 3

### Step 2: Determine how many men and women we need
We need to select:
- Men needed = 5
- Women needed = 2

### Step 3: Use the combination formula
The combination formula is given by:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

### Step 4: Calculate the combinations for men
We need to calculate the number of ways to choose 5 men from 7:
\[ C(7, 5) = \frac{7!}{5!(7 - 5)!} = \frac{7!}{5! \cdot 2!} \]
Calculating this:
\[ = \frac{7 \times 6 \times 5!}{5! \times (2 \times 1)} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

### Step 5: Calculate the combinations for women
Now, we calculate the number of ways to choose 2 women from 3:
\[ C(3, 2) = \frac{3!}{2!(3 - 2)!} = \frac{3!}{2! \cdot 1!} \]
Calculating this:
\[ = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = \frac{6}{2} = 3 \]

### Step 6: Multiply the combinations
Now, we multiply the number of ways to choose the men and the women:
\[ \text{Total combinations} = C(7, 5) \times C(3, 2) = 21 \times 3 = 63 \]

### Final Answer
Thus, the total number of ways to form a group of 5 men and 2 women from 7 men and 3 women is **63**.

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