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Problem 211

Combinatorics Difficulty 1.6 Multiple choice CEMC Gauss (Grade 8) · Canada · 2013

A family has 3 children. It is equally likely that a child is a boy as it is that a child is a girl. What is the probability that the 3 children are all girls?

Pick one

Official solution

Solution 1

Consider the set of five consecutive even numbers $8,10,12,14,16$ .

The mean of these five numbers is $\frac{8+10+12+14+16}{5}=\frac{60}{5}$ or $12$ .

If the five consecutive even numbers were smaller, their mean would be less than 12.

If the five consecutive even numbers were larger, their mean would be greater than 12.

Therefore, this is the set of five consecutive even numbers that we seek.
The mean of the smallest and largest of these five numbers is $\frac{8+16}{2}=\frac{24}{2}$ or 12.

Solution 2

The mean of five consecutive even numbers is the middle (third largest) number.

To see this, consider that the smallest of the five numbers is 4 less than the middle number, while the largest of the five numbers is 4 more than the middle number.

Thus, the mean of the smallest and largest numbers is the middle number.

Similarly, the second smallest of the five numbers is 2 less than the middle number, while the fourth largest of the five numbers is 2 more than the middle number.

Thus, the mean of these two numbers is also the middle number.

Since the mean of the five numbers is 12, then the middle number is 12.
Therefore, the mean of the largest and smallest of the five numbers is also 12.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.