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**Hint:**In this question we have the values of mean and the median given. We have to find the mode. So for this there is a relation between these three and it is given by $Mode = {\text{3 median - 2 mean}}$ . And is also known to be an empirical formula. So on substituting the values and equating we will get the value for the mode.

**Formula used:**

The relation between the mean, median and mode is given by

$Mode = {\text{3 median - 2 mean}}$

**Complete step-by-step answer:**First of all we will see the values given to us. Here in this question we have the value for the mean and median. And it is given by $14$ and $15$ respectively.

Now by using the relation, we have the relation as $Mode = {\text{3 median - 2 mean}}$

So on substituting the known values in the next line, we will get

$ \Rightarrow Mode = 3 \times 15 - 2 \times 14$

Now, on solving the multiplication we will get the solution as

$ \Rightarrow Mode = 45 - 28$

And therefore on solving the above differences, we will get the value for mode, and it will be

$ \Rightarrow Mode = 17$

Therefore, the value of the mode will be equal to $17$

**Hence, the option $\left( b \right)$ is correct.**

**Note:**The difference between the mean, median and mode in one line for each of the following will be given by as the mean is said to be as the average of a set of data. And the mode will be the most common number in the set of data. Whereas the median will be the middle of the numbers in the set. So in this way we can understand the basic difference between these three terms.

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