Siblings Problem - Combinatorics

[james.jt.wu]

Wanted to bump this question up mainly because this question is great for back-up method (thanks ron).

Method 3 to doing this problem - just list them!

First, total number of ways of picking 2 people out of 7 = (7*6)/2! = 21.

Next, the number of ways to pick a pair of sibling:

Use a letter to denote each person in the group:

A-B = first 2-sibling group
C-D= second 2-sibling group
E-F-G = 3-sibling group

So your pairs are:

AB
CD
EF
FG
EG

(remember here the positions are interchangeable... so AB and BA are the same pair)

5 total ways to pick two people who are siblings, so you probability of picking a sibling is 5/21.

Applying the 1-x rule: 1 - (5/21) = 16/21 probability of NOT sibling.

Bravo MGMAT - very nice question.

[mschwrtz]

Glad you liked it James. Maybe we should add Ron's note to the CAT explanation.

[WarrenGMAT]

4/7 you pick someone with 1 sibling. 3/7 you pick someone with 2 siblings.

If you pick a 1 sibling person, he has one sibling in the deck of 6 remaining people: 4/7 x 1/6 = 4/42 = 2/21

If you pick a 2 sibling person: 3/7 x 2/6 = 6/42 = 3/21

Add them: 5/21 says you get a sibling pair.

1 - 5/21 = 16/21 says you don't.

[jnelson0612]

Nice work everyone! Thank you to our contributors.

[gmatwork]

Can't we solve this problem as -

P(no siblings selected) = 1 - P( siblings selected)

P( siblings selected) = P(siblings from group 1) + P (Siblings from group 2) + P (siblings from group 3)

groups - are the three groups based on having two and three sibling pairs as said above in the posts

= 2C2/7C2 + 2C2/7C2 + 3C2/7C2
= 5/21

Then P(no siblings) = 1- 5/21 = 16/21

Please point out if I missed anything?

[gmatwork]

Can't we solve this problem as -

P(no siblings selected) = 1 - P( siblings selected)

P( siblings selected) = P(siblings from group 1) + P (Siblings from group 2) + P (siblings from group 3)

groups - are the three groups based on having two and three sibling pairs as said above in the posts

= 2C2/7C2 + 2C2/7C2 + 3C2/7C2
= 5/21

Then P(no siblings) = 1- 5/21 = 16/21

Please point out if I missed anything?

[tim]

Totally appropriate to do it this way, and you did it perfectly!