If the mean of a, b, c, d, e, f, ..., x, y, z (26 terms) is `beta,` then (`a - beta ) + (b-beta) + (c-beta) + ... + (x- beta + (y-beta) + (z-beta)` is equal to________

Options

A. 26
B. `beta`
C. 0
D. 25

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Mean**:
The mean (average) of a set of numbers is calculated by dividing the sum of all the numbers by the total count of the numbers. In this case, we have 26 terms: \( a, b, c, d, e, f, \ldots, x, y, z \).

2. **Given Information**:
We are given that the mean of these 26 terms is \( \beta \). Therefore, we can express this mathematically:
\[ \text{Mean} = \frac{a + b + c + d + e + f + \ldots + x + y + z}{26} = \beta \]

3. **Finding the Sum of the Terms**:
From the mean formula, we can rearrange it to find the sum of the terms:
\[ a + b + c + d + e + f + \ldots + x + y + z = 26 \beta \]

4. **Subtracting the Mean from Each Term**:
We need to calculate the expression:
\[ (a - \beta) + (b - \beta) + (c - \beta) + \ldots + (x - \beta) + (y - \beta) + (z - \beta) \]
This can be rewritten as:
\[ (a + b + c + \ldots + z) - 26\beta \]

5. **Substituting the Sum**:
We already found that the sum of all terms \( a + b + c + \ldots + z = 26\beta \). Substituting this into our expression gives:
\[ (26\beta) - 26\beta = 0 \]

6. **Final Result**:
Thus, the value of the expression \((a - \beta) + (b - \beta) + (c - \beta) + \ldots + (x - \beta) + (y - \beta) + (z - \beta)\) is equal to:
\[ \boxed{0} \]