Question: There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble is left in the jar?

1. 460
2. 490
3. 493
4. 445
5. 455

“There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble are left in the jar?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”.
To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

Total ways to select 8 marbles out of 7 + 5 = 12 is \(C^8_{12}\) ;

Ways to select 8 marbles so that zero red marbles is left in the jar is\(C^7_7*C^1_5\) ;

Ways to select 8 marbles so that zero blue marbles is left in the jar is \(C^5_5*C^3_7\);

Hence, the number of ways to select 8 marbles so that at least one red marble and at least one blue marble is left in the jar is \(C^8_{12}-(C^7_7*C^1_5+C^5_5*C^3_7)=495-(5+35)=455\)

Correct Answer: E

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