If my brother and I can invite 5 friends from a class of 15 to a party how many ways can we each pick 5 friends?

What's the probability 6 different friends are invited?

What's the prob my brother and I invite the same 5 people?

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- Feb 17th 2008, 09:24 PMmikeCombinations and Probability
If my brother and I can invite 5 friends from a class of 15 to a party how many ways can we each pick 5 friends?

What's the probability 6 different friends are invited?

What's the prob my brother and I invite the same 5 people? - Feb 18th 2008, 01:41 AMmr fantastic
Well, since this question's gone unanswered for a while, I'll stick my neck out here ....

(a) $\displaystyle {15 \choose 5} \times {15 \choose 5}$.

The reasoning: Each brother can choose 5 friends from 15 in $\displaystyle {15 \choose 5}$ ways. So the number of ways two brothers can choose is $\displaystyle {15 \choose 5} \times {15 \choose 5}$.

(b) $\displaystyle \frac{ {15 \choose 5} \times {5 \choose 4} \times {10 \choose 1}}{ {15 \choose 5} \times {15 \choose 5}} = \frac{{5 \choose 4} \times {10 \choose 1} }{{15 \choose 5}} = \frac{50}{ {15 \choose 5}} = .....$.

The reasoning:

numerator = number of ways the two brothers can choose 6 different friends: one brother chooses 5 friends from the 15, then the other brother chooses 4 friends from that 5 and 1 friend from the remaining 10.

denominator = number of ways the two brothers can choose 5 friends each = answer from part (a).

(c) $\displaystyle \frac{ {15 \choose 5} }{{15 \choose 5} \times {15 \choose 5}} = \frac{1}{{15 \choose 5}}$.

But since arrangements under restriction aren't my forte, you should probably wait for confirmation or refutation ...... - Feb 18th 2008, 07:27 AMmikeJust to check my answer...
(Worried) Is 15 C 5 = 3003?

- Feb 18th 2008, 09:59 AMmr fantastic
- Feb 18th 2008, 11:01 AMSoroban
Hello, mike!

I**totally**agree with*mr F*. . .

Quote:

If my brother and I can invite 5 friends from a class of 15 to a party,

how many ways can we each pick 5 friends?

Your brother has $\displaystyle {15\choose5}$ ways of picking 5 friends.

. . Therefore, the two of you have: .$\displaystyle {15\choose5} \times {15\choose5} \;=\;3003^2$ ways.

Quote:

What's the probability 6 different friends are invited?

__any__group of 5 friends . . . it doesn't matter.

Your brother must pick 4 of your 5 choices: .$\displaystyle {5\choose4} \:=\:5$ ways.

. . And he must pick one of the other ten friends: .$\displaystyle {10\choose1} \:=\:10$ ways.

. . Hence, he has $\displaystyle 5\times 10 \:=\:50$ ways to add a sixth friend.

Therefore, the probability is: .$\displaystyle \frac{50}{{15\choose5}} \;=\;\frac{50}{3003}$

Quote:

What's the prob my brother and I invite the same 5 people?

You can pick__any__group of 5 friends . . . it doesn't matter.

Your brother must pick the same 5 friends: .$\displaystyle 1$ way.

Therefore, the probability is: .$\displaystyle \frac{1}{{15\choose5}} \;=\;\frac{1}{3003}$