Combinations and Probability

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February 17th 2008, 09:24 PM
mike

Combinations 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?
February 18th 2008, 01:41 AM
mr fantastic

Quote:

Originally Posted by mike

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?

Well, since this question's gone unanswered for a while, I'll stick my neck out here ....

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

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

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

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

But since arrangements under restriction aren't my forte, you should probably wait for confirmation or refutation ......
February 18th 2008, 07:27 AM
mike

Just to check my answer...

(Worried) Is 15 C 5 = 3003?
February 18th 2008, 09:59 AM
mr fantastic

Quote:

Originally Posted by mike

(Worried) Is 15 C 5 = 3003?

Yes.
February 18th 2008, 11:01 AM
Soroban

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?

You have ${15\choose5}$ ways of picking 5 friends.
Your brother has ${15\choose5}$ ways of picking 5 friends.
. . Therefore, the two of you have: . ${15\choose5} \times {15\choose5} \;=\;3003^2$ ways.

Quote:

What's the probability 6 different friends are invited?

You can pick any group of 5 friends . . . it doesn't matter.

Your brother must pick 4 of your 5 choices: . ${5\choose4} \:=\:5$ ways.
. . And he must pick one of the other ten friends: . ${10\choose1} \:=\:10$ ways.
. . Hence, he has $5\times 10 \:=\:50$ ways to add a sixth friend.

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

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: . $1$ way.

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