Question

Fifteen people from different places came together for a family reunion. Each one of them had two gifts each for every other person. When the gifts were exchanged they hugged each other. What is the difference between the number of hugs and the number of gifts exchanged?

Hint:

For such problems: hugs = $\binom{n}{2}$, gifts = $n \times (n-1) \times \text{gifts per person}$.

Answer 1

Step 1: Count number of hugs.
Each of the 15 people hugs every other person exactly once.
Number of hugs = Combination of 15 taken 2: \[ \binom{15}{2} = \frac{15 \times 14}{2} = 105 \] Step 2: Count number of gifts.
Each person gives 2 gifts to each of the remaining 14 people.
Total gifts per person = $14 \times 2 = 28$.
For 15 people: $15 \times 28 = 420$.
Step 3: Difference.
\[ 420 - 105 = 315 \] Final Answer: \[ \boxed{315} \]