A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B. what is the least number that must have liked both products?

Explanation

To solve the problem, we will use the principle of inclusion-exclusion in set theory. Here's the step-by-step solution:

### Step 1: Understand the given data
We have:
- Total consumers surveyed, \( N = 1000 \)
- Consumers who like product A, \( N(A) = 720 \)
- Consumers who like product B, \( N(B) = 450 \)

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