The average of 15 different natural numbers is 13. The maximum value of the second largest of these nubers is

The average of the 15 values is given as 13.

Therefore, the total values will be $$15\times\ 13=195$$

As we want the maximum value of second largest number

So we will take the minimum value of first thirteen (15-2=13) natural numbers. 

As it is given in the question, we need to take different natural numbers, so the smallest 13 different natural number will be 

1, 2, 3, 4...13 Now when we sum the first 13 natural numbers, we get 

The formula for the summation of first n natural number is $$\dfrac{n\left(n+1\right)}{2}$$

For $$n=13$$, the sum will be equal to $$\dfrac{13\left(13+1\right)}{2}=91$$.

Therefore we will subtract 91 from the total value so 195-91=104

Now we need to find the maximum second-to-last number, so 104/2=52.

Now the two numbers will be 52-1=51 and 52+1=53, as the numbers should be different.

Therefore, the maximum value of the second largest number will be 51.

Hence option B is the correct answer.