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**Hint:**According to the question we have to find the total number of the binary operation formed by set having four elements. Here we use the formula for finding the number of binary operations with n elements ${n^{(n \times n)}}$.

**Complete step-by-step answer:**

In the question it is given a set having four elements {1,2,3,4}

The formula for finding the number of binary operations with n elements =${n^{(n \times n)}}$.

A set has four elements {1,2,3,4}

Hence the value of n = 4

So according to the question

$\therefore $The number of binary operations in a set with 4 elements

= ${4^{(4 \times 4)}}$

=${4^{16}}$

$\therefore $ The total number of binary operations in a set with 4 elements = ${4^{16}}$

**So, the correct answer is “Option C”.**

**Additional Information:**Binary operations on the set are the calculations that combine two elements of the set to produce another element of the same set.The elements of the set which produce another element of the set are termed as operands.Usually binary operations are denoted by special symbols such as $ \oplus , * ....$ etc. instead of using letters.Sets is a collection of well-defined objects which are different from each other.We usually represent sets by capital letters and elements of the sets by small letters.

**Note:**In these types of questions we have to remember that firstly the total number of the elements and secondly we have to remember that we have to write the solution in exponent form i.e ${4^{16}}$.

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