Difference between revisions of "2021 Fall AMC 12B Problems/Problem 23"
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What is the average number of pairs of consecutive integers in a randomly selected subset of <math>5</math> distinct integers chosen from the set <math>\{ 1, 2, 3, …, 30\}</math>? (For example the set <math>\{1, 17, 18, 19, 30\}</math> has <math>2</math> pairs of consecutive integers.) | What is the average number of pairs of consecutive integers in a randomly selected subset of <math>5</math> distinct integers chosen from the set <math>\{ 1, 2, 3, …, 30\}</math>? (For example the set <math>\{1, 17, 18, 19, 30\}</math> has <math>2</math> pairs of consecutive integers.) | ||
Revision as of 11:59, 24 November 2021
Problem
What is the average number of pairs of consecutive integers in a randomly selected subset of 
distinct integers chosen from the set 
? (For example the set 
has 
pairs of consecutive integers.)
Solution 1
There are 
possible pairs of consecutive integers, namely 
.
The probability that a certain pair of consecutive integers are in the integer subset is 
for the first number being chosen, multiplied by 
for the second number being chosen.
Therefore, by linearity of expectation, the expected number of pairs of consecutive integers in the 5-integer subset is ![\[\frac5{30}\cdot\frac4{29}\cdot29=\boxed{\textbf{(A)}\ \frac{2}{3}}.\]](http://latex.artofproblemsolving.com/2/d/6/2d6b066e23133d358c15269b0de91da3f171f0e5.png)
~kingofpineapplz
See Also
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 22 |
Followed by Problem 24 |
| 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 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
