Given:
Variance of first n natural numbers = 10
Variance of first m even natural numbers = 16

To find:
The value of m and n

Solution:
Let's start by understanding what variance means in statistics.

Variance:
Variance is a measure of how spread out the numbers in a dataset are. It quantifies the dispersion of a set of values. Mathematically, variance is the average of the squared differences from the mean.

Formula for Variance:
The formula for variance is given by:
Variance = (sum of (x - mean)^2) / n

where x is a data point, mean is the average of the data points, and n is the total number of data points.

Variance of first n natural numbers:
To find the variance of the first n natural numbers, we need to determine the sum of the squares of these numbers.

The sum of the squares of the first n natural numbers can be calculated using the formula for the sum of squares:
Sum of Squares = n * (n + 1) * (2n + 1) / 6

Using this formula, we can calculate the sum of squares for the first n natural numbers and substitute it into the variance formula.

Variance of first m even natural numbers:
Similarly, we can find the sum of squares of the first m even natural numbers and substitute it into the variance formula.

Equating the variances:
Since the variance of the first n natural numbers is given as 10 and the variance of the first m even natural numbers is given as 16, we can equate the two variances to find a relationship between n and m.

(sum of squares of the first n natural numbers) / n = 10
(sum of squares of the first m even natural numbers) / m = 16

Simplifying the equations:
By simplifying the equations, we get:
Sum of squares of the first n natural numbers = 10n
Sum of squares of the first m even natural numbers = 16m

Using the formulas for the sum of squares:
We can substitute the formulas for the sum of squares into the equations:
n * (n + 1) * (2n + 1) / 6 = 10n
m * (m + 1) * (2m + 1) / 6 = 16m

Solving the equations:
We can simplify the equations and solve for n and m:
n^3 + n^2 - 60n = 0
m^3 + m^2 - 48m = 0

By observing the equations, we can find that n = 10 satisfies the first equation, and m = 8 satisfies the second equation.

Therefore, the value of m * n = 8 * 10 = 80.

Conclusion:
The value of m * n is 80, which is not equal to 18. Therefore, the given answer of 18 is incorrect.