From 6 boys and 7 girls a committee of 5 is to be formed so as to incl...
To form a committee of 5 members, including at least one girl, we need to consider two cases:
1. When the committee has exactly one girl:
In this case, we need to select 1 girl from the 7 available girls and 4 boys from the 6 available boys. The number of ways to do this is given by:
7C1 * 6C4 = 7 * 15 = 105
2. When the committee has more than one girl:
In this case, we can have 2, 3, 4, or 5 girls in the committee. We need to consider each possibility separately and add up the results.
- When there are 2 girls in the committee:
We need to select 2 girls from the 7 available girls and 3 boys from the 6 available boys. The number of ways to do this is given by:
7C2 * 6C3 = 21 * 20 = 420
- When there are 3 girls in the committee:
We need to select 3 girls from the 7 available girls and 2 boys from the 6 available boys. The number of ways to do this is given by:
7C3 * 6C2 = 35 * 15 = 525
- When there are 4 girls in the committee:
We need to select 4 girls from the 7 available girls and 1 boy from the 6 available boys. The number of ways to do this is given by:
7C4 * 6C1 = 35 * 6 = 210
- When there are 5 girls in the committee:
We need to select all 5 girls from the 7 available girls. The number of ways to do this is given by:
7C5 = 21
Now, we can add up the results from each case to get the total number of ways to form the committee with at least one girl:
105 + 420 + 525 + 210 + 21 = 1281
Therefore, the correct answer is option D: 13C5 - 6C5, which is equal to 1281.