The Infinity Club consists of 5 boys and 6 girls. How many working groups of 5 people can be formed with:

a)no restrictions (my answer is 11C5)
b)all boys (is this correct 5C5)
c)all girls (answer 6C5)
d)more girls than boys (6C4x5C1 +6C3x5C2)
e)more boys than girls (5C4x6C1+5C3x6C2)

Let me explain how to get the answers for each part:

a) No restrictions:
To find the number of working groups with no restrictions, you'll use the combination formula. Since there are 11 people in total (5 boys + 6 girls), you'll calculate the number of combinations of 11 people taken 5 at a time. So the answer is 11C5.

b) All boys:
Since you want to form a group consisting of all boys, you choose all 5 boys from the available 5 boys. Using the combination formula, you calculate the number of combinations of 5 boys taken 5 at a time. So the answer is 5C5, which is equal to 1 (since there is only one way to choose all the boys).

c) All girls:
Similarly, for forming a group consisting of all girls, you choose all 6 girls from the available 6 girls. Using the combination formula, you calculate the number of combinations of 6 girls taken 5 at a time. So the answer is 6C5, which is equal to 6 (as there are 6 ways to choose all the girls).

d) More girls than boys:
To find the number of working groups with more girls than boys, you can break it down into two cases:
- Case 1: Choose 4 girls and 1 boy: You calculate the number of combinations of selecting 4 girls from the available 6 girls (6C4) and selecting 1 boy from the available 5 boys (5C1).
- Case 2: Choose 3 girls and 2 boys: You calculate the number of combinations of selecting 3 girls from the available 6 girls (6C3) and selecting 2 boys from the available 5 boys (5C2).

Add these two cases together to get the total number of groups. So the answer for part d) is (6C4 x 5C1) + (6C3 x 5C2).

e) More boys than girls:
Similar to part d), you'll break it down into two cases:
- Case 1: Choose 4 boys and 1 girl: You calculate the number of combinations of selecting 4 boys from the available 5 boys (5C4) and selecting 1 girl from the available 6 girls (6C1).
- Case 2: Choose 3 boys and 2 girls: You calculate the number of combinations of selecting 3 boys from the available 5 boys (5C3) and selecting 2 girls from the available 6 girls (6C2).

Add these two cases together to find the total number of groups. So the answer for part e) is (5C4 x 6C1) + (5C3 x 6C2).

Now you have all the explanations and answers for each part of the question.