If a, b and c are median, mode and range, respectively, of the data, 8, 5, 4, 3, 2, 7, 3, 10, 9, 17, 12, 3, 8, 4, then what is the value of (3a - 2b + c) ?

Options

A. 29
B. 30
C. 26
D. 27

Answer

To solve the problem, we need to find the median (a), mode (b), and range (c) of the given data set, and then calculate the expression \(3a - 2b + c\).

### Step 1: Arrange the Data in Ascending Order
The given data is:
\[ 8, 5, 4, 3, 2, 7, 3, 10, 9, 17, 12, 3, 8, 4 \]

Arranging this data in ascending order, we get:
\[ 2, 3, 3, 3, 4, 4, 5, 7, 8, 8, 9, 10, 12, 17 \]

### Step 2: Calculate the Median (a)
The number of data points (n) is 14, which is even. The median is calculated using the formula:
\[ \text{Median} = \frac{\text{n/2 th term} + \text{(n/2 + 1) th term}}{2} \]
Here, \(n/2 = 7\) and \((n/2 + 1) = 8\).

The 7th term is 5 and the 8th term is 7. Therefore,
\[ \text{Median} (a) = \frac{5 + 7}{2} = \frac{12}{2} = 6 \]

### Step 3: Calculate the Mode (b)
The mode is the number that appears most frequently in the data set. In our arranged data:
- 3 appears 3 times
- 4 appears 2 times
- All other numbers appear less frequently.

Thus, the mode (b) is:
\[ b = 3 \]

### Step 4: Calculate the Range (c)
The range is calculated as:
\[ \text{Range} (c) = \text{Highest value} - \text{Lowest value} \]
From our data, the highest value is 17 and the lowest value is 2.
\[ c = 17 - 2 = 15 \]

### Step 5: Calculate the Expression \(3a - 2b + c\)
Now we substitute the values of a, b, and c into the expression:
\[ 3a - 2b + c = 3(6) - 2(3) + 15 \]
Calculating this step by step:
\[ = 18 - 6 + 15 \]
\[ = 12 + 15 = 27 \]

### Final Answer
Thus, the value of \(3a - 2b + c\) is:
\[ \boxed{27} \]