A committee of 5 is to be formed from 9 ladies and 8 men. If the committee commands a lady majority, then the number of ways this can be done is

Options

A. 2352
B. 1008
C. 3360
D. 3486

Answer

To solve the problem of forming a committee of 5 members from 9 ladies and 8 men with a lady majority, we need to consider the different combinations of ladies and men that can form such a committee. A lady majority means that there must be more ladies than men in the committee. The possible combinations that satisfy this condition are:

1. 3 ladies and 2 men
2. 4 ladies and 1 man
3. 5 ladies and 0 men

We will calculate the number of ways to form the committee for each case and then sum them up.

### Step 1: Calculate the number of ways to choose 3 ladies and 2 men
The number of ways to choose 3 ladies from 9 is given by the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

So, the number of ways to choose 3 ladies from 9 is:
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

The number of ways to choose 2 men from 8 is:
\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2!6!} = \frac{8 \times 7}{2 \times 1} = 28 \]

Thus, the total ways for this case is:
\[ 84 \times 28 \]

### Step 2: Calculate the number of ways to choose 4 ladies and 1 man
The number of ways to choose 4 ladies from 9 is:
\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

The number of ways to choose 1 man from 8 is:
\[ \binom{8}{1} = 8 \]

Thus, the total ways for this case is:
\[ 126 \times 8 \]

### Step 3: Calculate the number of ways to choose 5 ladies and 0 men
The number of ways to choose 5 ladies from 9 is:
\[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

The number of ways to choose 0 men from 8 is:
\[ \binom{8}{0} = 1 \]

Thus, the total ways for this case is:
\[ 126 \times 1 \]

### Step 4: Sum up all the cases
Now, we sum the total ways from all three cases:
\[ (84 \times 28) + (126 \times 8) + (126 \times 1) \]

Calculating each term:
- \( 84 \times 28 = 2352 \)
- \( 126 \times 8 = 1008 \)
- \( 126 \times 1 = 126 \)

Now, adding them together:
\[ 2352 + 1008 + 126 = 3486 \]

### Final Answer
The total number of ways to form a committee of 5 members with a lady majority is **3486**.