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# If a + b + c + d = 4, then find 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 - b)(1 - a).

We will be reducing the given algebraic expression in the question.

## Answer: 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 - b)(1 - a) is zero (0).

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**Explanation:**

Given: a + b + c + d = 4

Take LCM of the denominators of 1/(1 - a)(1 - b)(1 - c), 1/(1 - b) (1 - c) (1 - d), 1/(1 - c)(1 - d)(1 - a) and 1/(1 - d)(1 - b)(1 - a) which is equal to (1 - a)(1 - b)(1 - c)(1 - d)

Hence, 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 - b)(1 - a) = [(1 - d) + (1 - a) + (1 - b) + (1 - c)] / [(1 - a)(1 - b)(1 - c)(1 - d)]

= [4 - (a + b + c + d)] / [(1 - a)(1 - b)(1 - c)(1 - d)] {Further, simplify the numerator}

= [4 - 4] / [(1 - a)(1 - b)(1 - c)(1 - d)] {since a + b + c + d = 4}

= 0 / [(1 - a)(1 - b)(1 - c)(1 - d)]

= 0

### Hence, the value of 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 - b)(1 - a) is zero (0), if a + b + c + d = 4.

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