To consider order or not to consider order help!

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KingKing: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Mon Oct 12, 2009 8:00 am

by KingKing » Mon Oct 12, 2009 8:58 am

Could someone please explain to me why my idea for answering the following question is not correct:

Suppose a box contains 10 balls of which 3 are red, 4 are white and 3 are green. A sample of 4 balls is selected at random without replacement. Find the probability of the event the sample contains 2 red balls

My process of thinking:
(ball 1) can be red in 3 ways
(ball 2 ) can be red in 2 ways
(ball 3) remaining colours in 7 ways
(ball 4) remaining colours in 6 ways

so desired outcomes can happen in 3x2x7x6 = 252 ways
How mayn possible outcomes in total = 10x9x8x7 = 5040 ways

so P(2 red balls in the set) = 252/5040

However the textbook give us the answer (3C2) * (7C2) / (10C4) = 3/10

Now I have read previous topics that discuss this confusion of when to choose "order matters" and "order does not matter"

https://www.beatthegmat.com/combination- ... t8924.html

Ian Stewart mentions that it does not matter which approach you take so as long as you are consistent in calculating numerator and denominator (ie calculating using permutations OR combinations) the difference should be by a constant .... could someone please clarify what I am doing wrong as I am consistent in my calculation

- Thanks

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Source: Beat The GMAT — Problem Solving | Mon Oct 12, 2009 8:58 am

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Re: To consider order or not to consider order help!

by Ian Stewart » Mon Oct 12, 2009 9:25 am

KingKing wrote:Could someone please explain to me why my idea for answering the following question is not correct:

Suppose a box contains 10 balls of which 3 are red, 4 are white and 3 are green. A sample of 4 balls is selected at random without replacement. Find the probability of the event the sample contains 2 red balls

My process of thinking:
(ball 1) can be red in 3 ways
(ball 2 ) can be red in 2 ways
(ball 3) remaining colours in 7 ways
(ball 4) remaining colours in 6 ways

so desired outcomes can happen in 3x2x7x6 = 252 ways
How mayn possible outcomes in total = 10x9x8x7 = 5040 ways

so P(2 red balls in the set) = 252/5040

However the textbook give us the answer (3C2) * (7C2) / (10C4) = 3/10

Now I have read previous topics that discuss this confusion of when to choose "order matters" and "order does not matter"

https://www.beatthegmat.com/combination- ... t8924.html

Ian Stewart mentions that it does not matter which approach you take so as long as you are consistent in calculating numerator and denominator (ie calculating using permutations OR combinations) the difference should be by a constant .... could someone please clarify what I am doing wrong as I am consistent in my calculation

- Thanks

You're answering a different question than that being asked - you've worked out the probability that the *first* two balls selected are red, and that the third and fourth are not red. That's not the only way to get two reds; there are 6 different ways to get two reds (RRnn, RnRn, RnnR, nRRn, nRnR, and nnRR), so if you multiply your answer by 6, you'll get the correct answer.

Alternatively, you can pretend order does not matter; we can choose the two reds in 3C2 ways, and the two non-reds in 7C2 ways, so can choose the four required balls in (3C2)(7C2) ways. Since there are 10C4 possible selections in total, we get the answer the book gave.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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