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# There are 3 red chips and 2 blue chips. If they form a certain color p

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Math Expert
Joined: 02 Sep 2009
Posts: 65764
There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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17 Sep 2019, 04:28
00:00

Difficulty:

25% (medium)

Question Stats:

70% (00:58) correct 30% (01:25) wrong based on 138 sessions

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There are 3 red chips and 2 blue chips. If they form a certain color pattern when arranged in a row, for example RBRRB, how many color patterns are possible?

A. 10
B. 12
C. 24
D. 60
E. 100

M09-20

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Joined: 02 Sep 2009
Posts: 65764
Re: There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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17 Sep 2019, 04:29
1
1
Bunuel wrote:
There are 3 red chips and 2 blue chips. If they form a certain color pattern when arranged in a row, for example RBRRB, how many color patterns are possible?

A. 10
B. 12
C. 24
D. 60
E. 100

M09-20

Official Explanation

THEORY

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:
$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

BACK TO QUESTION

Number of permutations of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

So, for our case the number of permutations of 5 letters BBRRR of which 2 B's and 3 R's are identical is $$\frac{5!}{2!*3!}=10$$.

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Re: There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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17 Sep 2019, 06:11
Bunuel wrote:
There are 3 red chips and 2 blue chips. If they form a certain color pattern when arranged in a row, for example RBRRB, how many color patterns are possible?

A. 10
B. 12
C. 24
D. 60
E. 100

M09-20

Number of color patterns =$$\frac{5!}{3!2!} = 10$$

IMO A
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There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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17 Sep 2019, 07:53
Bunuel wrote:
There are 3 red chips and 2 blue chips. If they form a certain color pattern when arranged in a row, for example RBRRB, how many color patterns are possible?

A. 10
B. 12
C. 24
D. 60
E. 100

M09-20

permutation ;
possible combinations ; 5!/3!*2! = 5*2 ; 10
IMO A
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Re: There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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17 Sep 2019, 10:32
total 5 chips,
so 5 chips can be arranged in 5! ways.
out of 5, 3 chips are of red color (similar) and 2 chips of blue color (similar).
so changing the places of similar chip will not create new pattern.
so total pattern = 5! / (3! *2!) = 10
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Re: There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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24 Sep 2019, 14:34
2
Bunuel wrote:
There are 3 red chips and 2 blue chips. If they form a certain color pattern when arranged in a row, for example RBRRB, how many color patterns are possible?

A. 10
B. 12
C. 24
D. 60
E. 100

M09-20

The number of possible patterns is 5! / (3!*2!) = (5 x 4) / 2 = 10.

Alternate Solution:

Note that if the 5 chips were all of different colors, then we would have 5! = 120 different arrangements. But because there are 3 identical red chips and 2 identical blue chips, we must reduce the number of possible arrangements, using the indistinguishable permutations formula, obtaining 5! / (3!*2!) = (5x4) / (2x1) = 10.

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Re: There are 3 red chips and 2 blue chips. If they form a certain color p  [#permalink]

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13 Jul 2020, 15:23
Can someone confirm the distinction between "order" vs "positioning" in the context of this problem and then broadly?

I define "order" and "positioning" as the following:
1) Order - where a distinct item or chip in this problem is placed in the arrangement or pattern in this problem
2) Positioning - where a simple item or chip in this problem is placed in the arrangement or pattern in this problem

I'll explain my italicizing of distinct and simple shortly but to differentiate the two, let me further qualify the chips in this problem as:

R1 - R2 - R3 - B1 - B2 ; where R1 is Red chip 1, R2 is Red chip 2, etc.

So for "Order" I've further qualified each of these chips as distinct by denoting each red and blue chip by a distinct number
While for "Positioning," in the context of this problem each simple qualification refers only to the color

But in the context of this problem, we only care about "Positioning" of each chip in the sense that the number of patterns we're looking for does not contemplate each chip being distinct only that the position of each simple chip (i.e. color) must be different

So do we determine we're concerned with "Positioning" or Combinations versus "Order" or Permutations in this problem simply from the words "color patterns" (with an emphasis on the color)?

Then if we removed the word "color" would we need to further qualify each simple chip as distinct and thus need to find permutations, not combinations?
Modified Question: There are 3 red chips and 2 blue chips. If they form a certain pattern when arranged in a row, how many patterns are possible?

To answer the above modified question, then would our answer be 5! or 120?

Thank you in advance for the help
Re: There are 3 red chips and 2 blue chips. If they form a certain color p   [#permalink] 13 Jul 2020, 15:23