how many x contain atleast one y

Question

Description

A group of 5 animals is to be chosen from 6 cats and 4 dogs.

Question

how many groups contain at least one dog?

Working Out

There are at most 4 dogs, so a group of 1 dog and 4 cats meet the condition as does a group of 2 dogs and 3 cats, 3 dogs, 2 cats and 4 dogs 1 cat.

So all I need to do is add the combinations together like so C(5,4) + C(5,3) + C(5,2) + (5,1) which equals 30 I believe.

Does that seem correct to you fokes? I have a feeling its wrong.

Your answer is not correct. It might be easier to calculate the total number of combinations and them subtract the ones that are not allowed, so composed of only cats. — Marc, Jun 04 '14 at 05:15
(1C,4D),(2C,3D),(3C,2D),(4C,1D) - Thus is ${6\choose1}{4\choose4}+{6\choose2}{4\choose3}+{6\choose3}{4\choose2}+{6\choose4}{4\choose1}$ Or else ${10\choose5}-{6\choose5}$ — Satish Ramanathan, Jun 04 '14 at 05:18
Thanks Satish, your explanation is what I think my book was trying to say :). — user133149, Jun 04 '14 at 05:36

score 2 · Answer 1 · answered Jun 04 '14 at 05:20

2

Total number of groups: $10\choose 5$

Number of groups with no dogs: $6\choose 5$

answered Jun 04 '14 at 05:20

poolpt

770

score 2 · Accepted Answer · answered Jun 04 '14 at 05:21

Choose 5 animals out of (6 + 4) animals. This can be done in $C(10, 5)$ ways. Now calculate the ways in which groups can be formed where the group members are only cats. This can be done in $C(6,5)$ ways.

So $C(10, 5) - C(6,5)$ are the ways in which a group can have atleast one dog.

how many x contain atleast one y

2 Answers2