Given are all possible combinations of 5 numbers from 1-35, all contained in combinations of 6 numbers (from 1-35). How many of the six number combinations contain 5 specific numbers?

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- Jan 22nd 2011, 06:46 AMaskjohnHow many combinations contain specific numbers?
Given are all possible combinations of 5 numbers from 1-35, all contained in combinations of 6 numbers (from 1-35). How many of the six number combinations contain 5 specific numbers?

- Jan 22nd 2011, 07:07 AMPlato
- Jan 22nd 2011, 07:17 AMaskjohn
Well, I'm a bit confused! That's what I thought at first too.

Since we are dealing with six number combinations, each one of which contains 6 possible combinations of five numbers, the total number of combinations of six containing all the combinations of five would be 324632 (35c5) / 6 = 54105 combinations of six; However, 35c6 would be 1623160.

So, if 30 were the answer to the first option, what would happen if we had all the combinations of 6 instead? Would it just be the same? - Jan 22nd 2011, 07:23 AMPlato
I just do not know what the OP means.

Is that an exact wording or is it a translation? - Jan 22nd 2011, 07:36 AMaskjohn
It's a translation. :)

Let me paraphrase.

We have all the possible combinations of five numbers from 1 to 35 (35c5) contained in combinations of six numbers.

for example: 1 2 3 4 5 6 (contains {1,2,3,4,5} {1,2,3,4,6} {1,2,3,5,6} {1,2,4,5,6} {1,3,4,5,6} {2,3,4,5,6})

What I want to know is how many of these -six- number combinations contain -five- specific numbers. Logically its 30. I'm confused however with the fact that every >1< combination of 6 contains 6 combinations of five!! :) - Jan 22nd 2011, 07:46 AMPlato
There are six combinations of six objects taken five at a time.

$\displaystyle \dbinom{6}{5}=\dfrac{6!}{5!\cdot 1!}=6$ - Jan 22nd 2011, 08:40 AMSoroban
Hello, askjohn!

The wording is confusing.

All that emphasis on subsets of 5 numbers . . .

Quote:

Given are all possible 6-number subsets of numbers from 1-35,

how many of these subsets contain 5 specific numbers?

What do you mean by "logically it's 30"? .Is that the correct answer?

Here's my reasoning . . .

The isway to get the five specific numbers.*one*

The sixth number can be any of the remaining 30 numbers.

Therefore, the answer is 30.

- Jan 22nd 2011, 08:45 AMaskjohn
You're right! It's so simple.. I just got confused by the subsets within the subsets! :) thanks!