The arithmetic mean of numbers a, b, c,d, e is M What is the value of (a - M)+ (b-M)+ (c-M) +(d-M)+(e-M)

Options

A. M
B. a+b+c+d+e
C. 0
D. 5M

Answer

To solve the problem, we need to find the value of the expression \((a - M) + (b - M) + (c - M) + (d - M) + (e - M)\) given that the arithmetic mean of the numbers \(a, b, c, d, e\) is \(M\).

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean \(M\) of the numbers \(a, b, c, d, e\) is given by the formula:
\[ M = \frac{a + b + c + d + e}{5} \]
This implies that:
\[ a + b + c + d + e = 5M \]

2. **Rewriting the Expression**:
We need to evaluate the expression:
\[ (a - M) + (b - M) + (c - M) + (d - M) + (e - M) \]
We can rewrite this expression by distributing the negative sign:
\[ = a - M + b - M + c - M + d - M + e - M \]
\[ = (a + b + c + d + e) - 5M \]

3. **Substituting the Sum**:
From step 1, we know that \(a + b + c + d + e = 5M\). We can substitute this into our expression:
\[ = 5M - 5M \]

4. **Simplifying the Expression**:
Now, simplifying the expression gives us:
\[ = 0 \]

### Final Answer:
Thus, the value of \((a - M) + (b - M) + (c - M) + (d - M) + (e - M)\) is:
\[ \boxed{0} \]