Question

For the Boolean expression \( Y = A B C + A' B C + A B' C + A' B' C \), the number of combinations for which the output \( Y = 1 \) is .............. .

Hint:

When simplifying Boolean expressions, try factoring out common terms and using Boolean algebra rules to make the expression easier to evaluate.

Answer 1

Correct Answer: 4

Step 1: Simplify the Boolean expression.
The given Boolean expression is \[ Y = A B C + A' B C + A B' C + A' B' C. \] We can factor out \( C \) from all terms, so the expression becomes \[ Y = C (A B + A' B + A B' + A' B'). \] Step 2: Simplify the expression inside the parentheses.
Notice that \[ A B + A' B + A B' + A' B' = (A + A') B + (A + A') B' = B + B' = 1. \] Thus, \[ Y = C \times 1 = C. \] Step 3: Find the combinations for which \( Y = 1 \).
Since \( Y = C \), the output will be 1 whenever \( C = 1 \). There are two possible combinations for \( A \) and \( B \) (00, 01, 10, or 11) when \( C = 1 \). Thus, the total number of combinations for which \( Y = 1 \) is \[ \boxed{4}. \]