From 6 boys and 7 girls a committee of 5 is to be formed so as to include atleast one girl. The number of ways this can be done is

Options

A. `""^(13)C_(4)`
B. `""^(6)C_(4).""^(7) C_(1)`
C. `7. ""^(6) C_(4)`
D. ` ""^(13)C_(5)-""^(6 )C_(1)`

Answer

To solve the problem of forming a committee of 5 members from 6 boys and 7 girls with the condition that at least one girl must be included, we can follow these steps:

### Step 1: Calculate the total ways to form a committee of 5 without restrictions
The total number of ways to select 5 members from 13 people (6 boys + 7 girls) is given by the combination formula:

\[ \text{Total ways} = \binom{13}{5} \]

### Step 2: Calculate the ways to form a committee with no girls
Next, we need to calculate the number of ways to form a committee of 5 members that consists only of boys. Since there are 6 boys, we can select all 5 members from these boys:

\[ \text{Ways with no girls} = \binom{6}{5} \]

### Step 3: Use the complement principle
To find the number of ways to form a committee that includes at least one girl, we subtract the number of all-boy committees from the total number of committees:

\[ \text{Ways with at least one girl} = \binom{13}{5} - \binom{6}{5} \]

### Step 4: Calculate the combinations
Now we will calculate the values of the combinations:

1. Calculate \(\binom{13}{5}\):
\[ \binom{13}{5} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287 \]

2. Calculate \(\binom{6}{5}\):
\[ \binom{6}{5} = 6 \]

### Step 5: Final calculation
Now we can substitute these values back into our equation:

\[ \text{Ways with at least one girl} = 1287 - 6 = 1281 \]

Thus, the total number of ways to form a committee of 5 members that includes at least one girl is **1281**.

### Final Answer
The number of ways to form the committee is **1281**.

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