If a+ b+c + d = 4 then the value of
`1/((1-a) (1-b) (1-c)) + 1/((1-b) (1-c) (1-d)) + 1/((1-c)(1-d)(1-a)) + 1/((1-d)(1-a) (1-b))` is

Options

A. 0
B. 1
C. 4
D. 1+ abcd

Answer

To solve the problem, we need to evaluate the expression given that \( a + b + c + d = 4 \). The expression is:

\[ E = \frac{1}{(1-a)(1-b)(1-c)} + \frac{1}{(1-b)(1-c)(1-d)} + \frac{1}{(1-c)(1-d)(1-a)} + \frac{1}{(1-d)(1-a)(1-b)} \]

### Step 1: Choose values for \( a, b, c, d \)

Since \( a + b + c + d = 4 \), we can choose values that satisfy this equation. A simple choice is:

- \( a = 0 \)
- \( b = 0 \)
- \( c = 2 \)
- \( d = 2 \)

This gives us:

\[ 0 + 0 + 2 + 2 = 4 \]

### Step 2: Substitute the values into the expression

Now we substitute \( a, b, c, d \) into the expression \( E \):

1. For the first term:
\[ (1-a)(1-b)(1-c) = (1-0)(1-0)(1-2) = 1 \cdot 1 \cdot (-1) = -1 \]
Thus, the first term becomes:
\[ \frac{1}{-1} = -1 \]

2. For the second term:
\[ (1-b)(1-c)(1-d) = (1-0)(1-2)(1-2) = 1 \cdot (-1) \cdot (-1) = 1 \]
Thus, the second term becomes:
\[ \frac{1}{1} = 1 \]

3. For the third term:
\[ (1-c)(1-d)(1-a) = (1-2)(1-2)(1-0) = (-1)(-1)(1) = 1 \]
Thus, the third term becomes:
\[ \frac{1}{1} = 1 \]

4. For the fourth term:
\[ (1-d)(1-a)(1-b) = (1-2)(1-0)(1-0) = (-1)(1)(1) = -1 \]
Thus, the fourth term becomes:
\[ \frac{1}{-1} = -1 \]

### Step 3: Combine the results

Now we combine all the terms:

\[ E = -1 + 1 + 1 - 1 = 0 \]

### Conclusion

Thus, the value of the expression is:

\[ \boxed{0} \]