1989 AHSME Problems/Problem 30
Contents
[hide]Problem
Suppose that 7 boys and 13 girls line up in a row. Let be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row we have that . The average value of (if all possible orders of these 20 people are considered) is closest to
Solution 1
We approach this problem using Linearity of Expectation. Consider a pair of two people standing next to each other. Ignoring all other people, the probability that a boy is standing on the left position and a girl is standing on the right position is . Similarly, if a girl is standing on the left position and a boy is standing on the right position the probability is also . Thus, the total probability of the two people being one boy and one girl is .\ \ There are a total of 19 different adjacent pairs, so by Linearity of Expectation, we have that the expected value of is .
Solution 2
Suppose that the class tried every configuration. Boy and girl would stand next to each other in different orders, in different positions, times each. Summing over all gives , so the average value of is .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.