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Alejandra is designing a game of chance. For one part of the game, Updated on: 30 Apr 2024, 14:00
Originally posted by GMATCoachBen on 30 Apr 2024, 13:58.
Last edited by GMATCoachBen on 30 Apr 2024, 14:00, edited 2 times in total.
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1: 1 less than 2: twice
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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.

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1 2
2 more than
1 more than
1 less than
2 less than
half
twice
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Alejandra is designing a game of chance. For one part of the game, 30 Apr 2024, 13:59
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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
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Alejandra is designing a game of chance. For one part of the game, 30 Apr 2024, 14:17
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GMATCoachBen
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.

 ­
Show more
­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.










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Alejandra is designing a game of chance. For one part of the game, 14 Jun 2024, 23:17
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B c 2 * R c 1 = 2 * R c 2 * B c 1
B^2 = B(2R -1)
B = 2r -1

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Alejandra is designing a game of chance. For one part of the game, 19 Jun 2024, 10:14
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GMATCoachBen
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.

 ­
Show more
­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.

BC2*RC1 = 2*BC1*RC2 
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
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Alejandra is designing a game of chance. For one part of the game, 14 Jul 2025, 05:05
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