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GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 57

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If $n$ is a positive integer, the notation $n!$ (read " $n$ factorial") is used to represent the product of the integers from $1$ to $n$ inclusive. For example, $5 !=1 \times 2 \times 3 \times 4 \times 5=120$. Which of the following is equal to a perfect square?

  1. $\dfrac{(20 !)(19 !)}{2}$
  2. $\dfrac{(20 !)(19 !)}{3}$
  3. $\dfrac{(20 !)(19 !)}{4}$
  4. $\dfrac{(20 !)(19 !)}{5}$
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Answer:Option D

20! = 20*19*18*17*….

19!=19*18*17*….

We just take the common numbers and then only 20 will be remaining i..e.

20 * 19^2 * 18^2 * …..

If 20/5=4 and 4 can be represented as 2^2.Then all the numbers will be in squares.And that number can be considered as perfect number
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