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There are 12 cans in the refrigerator. 7 of them are red and 5 of them are blue. In how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator.

There are 12 cans in the refrigerator. 7 of them are red and 5 of them are blue. In how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator.

(A) 460 (B) 490 (C) 493 (D) 455 (E) 445

'D' - 455

Ways to pick 4 cans so that at least one red and at least one blue cans to remain the refrigerator = total ways to pick 4 can out of 12 - ways to pick 4 red out of 7 red - ways to pick 4 blue out of 5 blue

\(12C4 - 7C4 - 5C4 = 495 - 35 -5 = 455\)

Last edited by swatirpr on 20 Nov 2009, 07:41, edited 1 time in total.

Ways to pick 4 cans so that at least one red and at least one blue cans to remain the refrigerator = total ways to pick 4 can out of 12 - ways to pick 4 red out of 7 red - ways to pick 4 blue out of 5 blue

\(12C4 - 7C4 - 5C4 = 495 - 35 -5 = 455\)

Sounds logical. Question: Do we know that we always have 7 red cans to choose from?? Out of these 7..4 can be in the 'chosen 8 cans'...so we are left with only 3 red cans to choose from.

Total number of ways to selecting 8 cans from 12 cans MINUS [ number of ways of leaving no red can at all in choosing 8 cans PLUS number of ways of leaving no blue can at all in choosing 8 cans]

12C8 - [ (7C7*5C1) + (5C5*7C3)] = 495 - 40 = 455.

D is the answer, what is the OA? _________________

Total number of ways to selecting 8 cans from 12 cans MINUS [ number of ways of leaving no red can at all in choosing 8 cans PLUS number of ways of leaving no blue can at all in choosing 8 cans]

12C8 - [ (7C7*5C1) + (5C5*7C3)] = 495 - 40 = 455.

D is the answer, what is the OA?

No need to postpone the correct answer. It's my question, so only my solution is available. And it's exactly the one Barney proposed. Perfect logic +1.

I considered all cans are identical in respective color group, ensuring at least 1R and 1B by considering these two groups ( 6R, 4B )

Case 1: 6R 2B ( blue will naturally take empty space ) thus we should calculate only 8C6 to mark possible slots on 8 available turns i.e = 28 Case 2: 5R 3B similarly 8C5 = 56. Case 3: 4R 4B similarly 8C4 = 70. <I can not further decrease R below 4 as it will take 3R 5B which violates the criterion>

E.g R _ R _ R _ R _ or RR_ _ R_ _ or _ _ R_R_RR (we can see empty slots are available for blue by default) thus combination for Red cans on 8 slots is sufficient, as per my understanding.

But this approach gives me result as 28 + 56 + 70 = 154. Which is not there in the options, is my approach right ? _________________

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I considered all cans are identical in respective color group, ensuring at least 1R and 1B by considering these two groups ( 6R, 4B )

Case 1: 6R 2B ( blue will naturally take empty space ) thus we should calculate only 8C6 to mark possible slots on 8 available turns i.e = 28 Case 2: 5R 3B similarly 8C5 = 56. Case 3: 4R 4B similarly 8C4 = 70. <I can not further decrease R below 4 as it will take 3R 5B which violates the criterion>

E.g R _ R _ R _ R _ or RR_ _ R_ _ or _ _ R_R_RR (we can see empty slots are available for blue by default) thus combination for Red cans on 8 slots is sufficient, as per my understanding.

But this approach gives me result as 28 + 56 + 70 = 154. Which is not there in the options, is my approach right ?

Obviously not.

There are 12 cans in the refrigerator. 7 of them are red and 5 of them are blue. In how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator. (A) 460 (B) 490 (C) 493 (D) 455 (E) 445

Total ways to select 8 cans out of 7+5=12 is \(C^8_{12}\); Ways to select 8 cans so that zero red cans are left is \(C^7_7*C^1_5\); Ways to select 8 cans so that zero blue cans are left is \(C^5_5*C^3_7\);

Hence ways to select 8 cans so that at least one red can and at least one blue can to remain is \(C^8_{12}-(C^7_7*C^1_5+C^5_5*C^3_7)=495-(5+35)=455\).

I don't understand how did you guys get to the numerical numbers I mean I know it might be a basic question but i don't see the connection between the values

Bunnel I don't get it. Question says ...that at least one red and at least one blue cans to remain the refrigerator...

I think D would be answer if question were ...that at least one red OR at least one blue cans to remain the refrigerator...notice OR.

please enlighten me if I am going to wrong way.

Posted from my mobile device

The question asks: in how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator.

So, we can remove: 6 red, 2 blue (1 red, 3 blue are left) --> \(C^2_5*C^6_7=70\); 5 red, 3 blue (2 red, 3 blue are left) --> \(C^3_5*C^5_7=210\); 4 red, 4 blue (3 red, 1 blue are left) --> \(C^4_5*C^4_7=175\);

70+210+175=455.

The way it's olved in my post is different: (total)-(restriction).

and one thing, ways in which at least one blue OR at least one red remains would be 12c8 . Isn't it ?

Posted from my mobile device

Yes. In this case no matter which 8 cans we remove from 12, there still will be at least 1 red or at least one blue can remaining in the ref: 12C8=495.

In this case we could also go the long way: 3 cases from my previous post + 2 more cases: 7 red, 1 blue (0 red, 4 blue are left) --> \(C^1_5*C^7_7=5\); 3 red, 5 blue (4 red, 0 blue are left) --> \(C^5_5*C^3_7=35\);

Total = 455(from my previous post) + 35 + 5 = 495.

Re: There are 12 cans in the refrigerator. 7 of them are red and [#permalink]

Show Tags

18 Sep 2013, 01:51

Bunuel wrote:

There are 12 cans in the refrigerator. 7 of them are red and 5 of them are blue. In how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator.

(A) 460 (B) 490 (C) 493 (D) 455 (E) 445

1. Selection has to be made from each of two distinct groups, being red cans and blue cans

2. n1=7, n2=5

3. r1= 6 and r2=2 or r1=5 and r2=3 or r1=4 and r3=4,

each of which satisfies that there is at least one red can and one blue can left.

The total number of ways = 7C6*5C2 + 7C5*5C3 + 7C4*5C4 = 455 _________________

Re: There are 12 cans in the refrigerator. 7 of them are red and [#permalink]

Show Tags

30 Dec 2013, 08:12

Bunuel wrote:

There are 12 cans in the refrigerator. 7 of them are red and 5 of them are blue. In how many ways we can remove 8 cans so that at least one red and at least one blue cans to remain the refrigerator.

(A) 460 (B) 490 (C) 493 (D) 455 (E) 445

7C6*5C3 + 7C4*5C3 + 7C4*5C4 = 455

Answer is D

Hope it helps! Cheers!

PS. I suggest listing range of values of each can just to visualize problem better Then just pick the values that add up to 8 Namely, (2+6), (3+5) and (4+4)

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