Question: Ten coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?

1. \(2^9\)
2. \(2^{10}\)
3. \(3*2^8\)
4. \(3*2^9\)
5. \(3*2^{10}\)

Correct Answer: A

Solution and Explanation
Approach Solution 1:
Please note this question is about the outcome not arrangement, one of the most common areas of confusion.

Apart from 3rd toss which needs to be Heads all other outcomes are free and independent of each other.
Total outcome for remaining 9 toss =\(2^9\)

3rd toss outcome = 1 way
Hence total outcome = \(2^9\)* 1 =\(2^9\)

Approach Solution (2):

First point: If there are no restrictions, outcomes = 2*2*2… 10 times =\(2^{10}\)
Each coin can have two outcomes: head or tail, so 2*2*2…
Second point: the restrictions, third to be head only…

Since the third coin has 2 outcomes, half of the total will have head and half tail…

So outcomes = \(\frac{2^{10}}{2}=2^9\)

Approach Solution 3:
When a coin is tossed once, there are two outcomes. It can turn up a head or a tail.
When 10 coins are tossed simultaneously, the total number of outcomes = 210
Out of these, if the third coin has to turn up a head, then the number of possibilities for the third coin is only 1 as the outcome is fixed as head.

Therefore, the remaining 9 coins can turn up either a head or a tail = 29

“Ten coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?”- 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.

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