At a Certain Laboratory, Chemical Substances are Identified by an Unordered Combination of Three Different Colours GMAT Problem Solving

Question: At a certain laboratory, chemical substances are identified by an unordered combination of three different colours. If no chemical may be assigned the same three colours as any other, what is the maximum number of substances that can be identified using seven colours?

A) 21
B) 35
C) 105
D) 135
E) 210

“At a Certain Laboratory, Chemical Substances are Identified by an Unordered Combination of Three Different Colours”- 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 number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

The model answer given has only 1 approach of using the combination formula.

Explanation: The given situation states that there are three different colours of chemical substances which have been arranged in an unordered combination. What is to be evaluated from here is the number of substances that can be identified from the combination of seven colours. This given that no other chemical may be assigned to the existing three colours. 

The main problem to be solved is to find the number of ways in which the seven colours can be combined together in groups of three. Clearly this is a problem of combination which needs to be solved. 

The formula which is typically used for solving a combination problem is - 

\(nCr=\frac {n!}{r!(n-r)!}\)

Here, n is the total number of objects present in the set and r represents the number that needs to be combined at one time. Accordingly, in the current scenario, n is equal to 7 and r equal to the combination of 3 colours that is to be used to identify a chemical. This can be evaluated as follows: 

\(\frac{7!}{3!(7-3)!}=\frac{7!}{3!4!}\)

\(=\frac{7*6*5}{3*2*1}\)

This is gained when the 4*3*2*1 is cancelled from the top and the bottom. From this above equation the answer gained can be evaluated as- 

210/6 = 70/2= 35

From the above evaluation, it can be stated that 35 is the total number of ways in which seven colours can be combined into the groups of 3. Hence, the answer gained from this combination problem is option B. 

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