A team of five is randomly chosen from six girls and seven boys. How many different ways can the team be chosen if there must be more girls than boys in the team?
To find the number of different ways a team can be chosen with more girls than boys, we can consider the different possibilities and calculate them separately.
First, let's consider the case where all members of the team are girls. Since there are six girls available, we can choose all five girls from the six available in 6 choose 5 ways, which can be expressed as "6C5" or mathematically calculated as follows:
6C5 = 6! / (5! * (6-5)!) = 6
Next, let's consider the case where four girls and one boy are chosen. There are two sub-cases to consider here: either one of the boys is chosen or the other boy is chosen.
Sub-case 1: One of the boys is chosen:
- Select 1 boy from 7 available boys in 7 choose 1 ways
- Select 4 girls from 6 available girls in 6 choose 4 ways
- Multiply these two selections together: 7C1 * 6C4 = 7 * 15 = 105
Sub-case 2: The other boy is chosen:
- Select 1 boy from 7 available boys in 7 choose 1 ways
- Select 4 girls from 6 available girls in 6 choose 4 ways
- Multiply these two selections together: 7C1 * 6C4 = 7 * 15 = 105
Finally, let's consider the case where three girls and two boys are chosen. Similar to the previous case, there are two sub-cases to consider.
Sub-case 1: Two of the boys are chosen:
- Select 2 boys from 7 available boys in 7 choose 2 ways
- Select 3 girls from 6 available girls in 6 choose 3 ways
- Multiply these two selections together: 7C2 * 6C3 = 21 * 20 = 420
Sub-case 2: One boy and one of the girls chosen:
- Select 1 boy from 7 available boys in 7 choose 1 ways
- Select 1 girl from 6 available girls in 6 choose 1 ways
- Select 2 girls from the remaining 5 available girls in 5 choose 2 ways
- Multiply these three selections together: 7C1 * 6C1 * 5C2 = 7 * 6 * 10 = 420
Now, to find the total number of different ways the team can be chosen, we need to sum up the possibilities from all the cases.
Total number of different ways = Case 1 + Case 2 (Sub-case 1 + Sub-case 2) + Case 3 (Sub-case 1 + Sub-case 2)
= 6 + 105 + 105 + 420 + 420
= 1056
Therefore, there are 1056 different ways to choose a team of five members with more girls than boys.