Sample Numeric Answer Question

There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feets?

Question

Sample Numeric Answer Question

There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feets?

BlackTom AI · Accepted Answer

Using the inclusion–exclusion principle for two overlapping groups, we can count how many people wear at least one of socks or shoes, then subtract from the total to find those barefoot.

1. Relevant knowledge points and concepts
- Venn diagram / set notation for two sets: S (people wearing socks) and H (people wearing shoes).
- Inclusion–exclusion principle for two sets: |S ∪ H| = |S| + |H| − |S ∩ H|.
- “Bare feet” means neither socks nor shoes: count = total people − |S ∪ H|.

2. Complete reasoning process step by step
- Total people = 12.
- Number wearing socks |S| = 6.
- Number wearing shoes |H| = 4.
- Number wearing both |S ∩ H| = 3.
- Use inclusion–exclusion to find number wearing at least one (socks or shoes): |S ∪ H| = |S| + |H| − |S ∩ H|.

3. Calculations with intermediate steps
- Compute |S| + |H| = 6 + 4 = 10.
- Subtract the overlap: 10 − |S ∩ H| = 10 − 3 = 7.
- So 7 people wear at least one of socks or shoes.
- Barefoot count = total − |S ∪ H| = 12 − 7 = 5.

4. Why the correct answer is correct
- People wearing both were counted twice when simply adding socks and shoes (once in each group). Subtracting the overlap corrects this double-count. After correction we get the exact number wearing some footwear (socks or shoes). The remainder of the group are those wearing neither, i.e., barefoot. That remainder is 5.

5. Why other answers (common mistakes) would be incorrect
- Incorrectly adding without subtracting overlap: 6 + 4 = 10 would overcount the 3 people who are in both groups.
- Incorrectly subtracting instead of using inclusion–exclusion (e.g., 12 − 6 − 4 = 2) ignores that the 3 both-wearing people were subtracted twice.
- Using correct set arithmetic as above avoids these mistakes.

6. Conclusion
The number of people barefoot is 5, which matches the provided answer.

Sample Numeric Answer Question There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feets?数值题

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Sample Numeric Answer Question There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feet?

Sample Numeric Answer Question There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feet?

Sample Numeric Answer Question There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feets?

Sample Numeric Answer Question There are 12 people in a room. Six people are wearing socks and four people are wearing shoes; three people are wearing both. How many are in bare feet?

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