Alejandra is designing a game of chance. For one part of the game,

Question

Alejandra is designing a game of chance. For one part of the game, a player is to randomly choose 3 marbles, without replacement, from a box containing B blue marbles, R red marbles, and no other marbles. Alejandra correctly determined the positive integers B and R so that the number of possible selections in which 2 blue marbles and 1 red marble are chosen is twice the number of possible selections in which 2 red marbles and 1 blue marble are chosen.

The positive integers B and R that Alejandra determined must be such that B is the number that is ___ 1 ___ the number that is ___ 2 ___ R.

Based on the information provided, select for  1  and for  2  the options that create the most accurate statement. Make only two selections, one in each column.

GMATCoachBen · Accepted Answer

We need to translate this into an equation:

"the number of possible selections in which 2 blue marbles and 1 red marble are chosen is twice the number of possible selections in which 2 red marbles and 1 blue marble are chosen."

For the 2 blue and 1 red, note that "And" means "multiply".  We are choosing 2 blue out of B total blue marbles, and 1 red out of R total red marbles. Let's use this timesaving formula for combinations, when we are choosing 2 out of N: (NC2) =  \frac{N(N-1) }{ 2} . (This formula for "pairs" can come up in other contexts, such as handshakes or games played between 2 teams.)

(BC2) * (RC1) =  \frac{B(B-1) }{ 2}   * R

For the 2 red and 1 blue, we use the same method:

(RC2) * (BC1) =  \frac{R(R-1) }{ 2}   * B

Then, let's set up our full equation. It says that 2 blue and 1 red is twice the number for 2 red and 1 blue:

\frac{B(B - 1) }{ 2}   * R = 2 *  \frac{R(R - 1) }{ 2}   * B

We simplify, by crossing off the B and R on both sides, and 2 on the bottom on both sides:

(B - 1)= 2(R - 1)

B = 2R - 1

Finally, we read carefully and double-check that we are filling in the blanks correctly, so that it matches our equation:

"𝐵 is the number that is ___ 1 ___ the number that is ___ 2 ___ 𝑅."

B is  1 less than  the number that is  twice  R.

GMATCoachBen · Answer

This one was on Exam 4. There is another similar version posted here:  https://gmatclub.com/forum/alejandra-is-designing-a-game-of-chance-for-one-part-of-the-game-422148.html#p3305806

sayan640 · Answer

B c 2 * R c 1 = 2 * R c 2 * B c 1
B^2 = B(2R -1)
B = 2r -1

Ans

GMATinsight · Answer

Alejandra correctly determined the positive integers 𝐵 and 𝑅 so that the number of possible selections in which 2 blue marbles and 1 red marble are chosen is twice the number of possible selections in which 2 red marbles and 1 blue marble are chosen. B C 2*R C 1 = 2* B C 1*R C 2 B(B-1)*R/2 = 2*R(R-1)*B/2 i.e. B-1 = 2(R-1) i.e. B = 2R-1 So, B is the number that is 1 less than twice the number R. hence the answer --- GMATinsight | www.GMATinsight.com GMAT FOCUS PREP Expert Youtube: www.Youtube.com/GMATinsight (1100+ Quant and DI prep videos FREE)

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