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 (my answer 6C5)
d)more girls than boys (answer 6C4x5C1 +6C3x5C2)?
e)more boys than girls (5C4x6C1+5C3x6C2)?
Thanks in advance
a)no restrictions (my answer is 11C5) -- correct
b)all boys (is this correct 5C5) -- correct
c)all girls (my answer 6C5) -- correct
d)more girls than boys (answer 6C4x5C1 +6C3x5C2)?
---- more girls than boys ---> 3G,2B - 4G,1B - 5G
= C(6,3)xC(5,2) + C(6,4)xC(5,1) + C(6,5)
e)more boys than girls --- wrong
--- do it the same way as I showed you in d)
e)5C3x6C2+5C4x6C1+5C5 is this correct
Thank you very much
a) To find the number of working groups with no restrictions, you can use the combination formula. In this case, you have 11 people to choose from, and you want to select a group of 5. So the answer is:
11C5 = 11! / (5! * (11-5)!) = 462
b) To find the number of working groups consisting only of boys, you have 5 boys to choose from, and you want to select a group of 5. Since there are no restrictions, you can simply choose all 5 boys. So the answer is:
5C5 = 1
c) To find the number of working groups consisting only of girls, you have 6 girls to choose from, and you want to select a group of 5. Since there are no restrictions, you can simply choose all 5 girls. So the answer is:
6C5 = 6
d) To find the number of working groups with more girls than boys, you can consider the combinations of choosing 4 girls and 1 boy, as well as 3 girls and 2 boys. Then add them up to get the total number of groups.
First, select 4 girls out of 6:
6C4 = 6! / (4! * (6-4)!) = 15
Then select 1 boy out of 5:
5C1 = 5
Multiply the two results together to get the number of groups with 4 girls and 1 boy.
15 * 5 = 75
Next, select 3 girls out of 6:
6C3 = 6! / (3! * (6-3)!) = 20
Then select 2 boys out of 5:
5C2 = 10.
Multiply the two results together to get the number of groups with 3 girls and 2 boys.
20 * 10 = 200
Finally, add the two results together to get the total number of groups with more girls than boys:
75 + 200 = 275
d) Similarly, to find the number of working groups with more boys than girls, you can consider the combinations of choosing 4 boys and 1 girl, as well as 3 boys and 2 girls. Then add them up to get the total number of groups.
First, select 4 boys out of 5:
5C4 = 5! / (4! * (5-4)!) = 5
Then select 1 girl out of 6:
6C1 = 6
Multiply the two results together to get the number of groups with 4 boys and 1 girl.
5 * 6 = 30
Next, select 3 boys out of 5:
5C3 = 5! / (3! * (5-3)!) = 10
Then select 2 girls out of 6:
6C2 = 15.
Multiply the two results together to get the number of groups with 3 boys and 2 girls.
10 * 15 = 150
Finally, add the two results together to get the total number of groups with more boys than girls:
30 + 150 = 180
So the correct answers are:
a) 462
b) 1
c) 6
d) 275
e) 180