Calculate Mean of Set S

The given set is S = {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}. The total number of elements is 10.

First, find the sum of all elements in the set:

Sum = $1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 4 + 4 = 30$.

The mean is calculated by dividing the sum by the number of elements:

Mean = $\frac{\text{Sum of elements}}{\text{Number of elements}} = \frac{30}{10} = 3$.

Determine Mode of Set S

The mode is the value that appears most frequently in the data set.

Count the frequency of each number in S:

• 1 appears 1 time
• 2 appears 2 times
• 3 appears 3 times
• 4 appears 4 times

The number 4 has the highest frequency (4 times). Therefore, the mode is 4.

Find Median of Set S

The set S is already sorted: {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}.

Since there are 10 elements (an even number), the median is the average of the two middle elements, which are the 5th and 6th elements.

The 5th element is 3.

The 6th element is 3.

Median = $\frac{5\text{th element} + 6\text{th element}}{2} = \frac{3 + 3}{2} = \frac{6}{2} = 3$.

Evaluate Expression: $4 \times \text{mean} + 2 \times \text{mode} - 8 \times \text{median}$

Now substitute the calculated values of mean, mode, and median into the expression:

• Mean = 3
• Mode = 4
• Median = 3

Expression = $4 \times (3) + 2 \times (4) - 8 \times (3)$

Perform the calculations:

Expression = $12 + 8 - 24$

Expression = $20 - 24 = -4$.

The value of the expression $4 \times \text{mean} + 2 \times \text{mode} - 8 \times \text{median}$ is -4.