Question

In a group of children, each child gives a gift to every other. If the number of gifts is 132, then the number of children is:

• A. 11
• B. 12
• C. 13
• D. 10

Hint:

When dealing with gift exchange problems, use combinations to calculate the total number of ways each child can give a gift to every other child.

Answer 1

The Correct Option is B

Let the number of children be \( n \). 
Step 1: Each child gives a gift to every other child, so the total number of gifts is given by the combination formula: \[ \binom{n}{2} = \frac{n(n-1)}{2} \] 
Step 2: Set this equal to the total number of gifts: \[ \frac{n(n-1)}{2} = 132 \] 
Step 3: Multiply both sides by 2: \[ n(n - 1) = 264 \] 
Step 4: Solve the quadratic equation: \[ n^2 - n - 264 = 0 \] Using the quadratic formula: \[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-264)}}{2(1)} \] \[ n = \frac{1 \pm \sqrt{1 + 1056}}{2} = \frac{1 \pm \sqrt{1057}}{2} \] Approximating the square root of 1057, we get: \[ n \approx 12 \] Thus, the number of children is 12.