Random Number Multiplication
Three integers, a , b , and c are chosen randomly from the following ranges:
1 ≤ a 2 ≤ b 3 ≤ c ≤ 5 ≤ 6 ≤ 7
For all possible combinations of a , b , and c , what is the mean average of the values given by a b c ?
The answer is 60.
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2 solutions
Factoring the sum of all possible values is really the best solution here! Also, it can be seen that the average of the product is the same as the product of the averages of the individual factors.
= = = 1 2 5 1 ( 1 + 2 + 3 + 4 + 5 ) ( 2 + 3 + 4 + 5 + 6 ) ( 3 + 4 + 5 + 6 + 7 ) 5 1 + 2 + 3 + 4 + 5 ⋅ 6 2 + 3 + 4 + 5 + 5 ⋅ 5 3 + 4 + 5 + 6 + 7 3 ⋅ 4 ⋅ 5 6 0
<?php
$vals = array();
for($a=1;$a<=5;$a++){
for($b=2;$b<=6;$b++){
for($c=3;$c<=7;$c++){
$vals[] = $a * $b * $c;
}
}
}
echo array_sum($vals)/count($vals); //60
?>
By expanding, it can be seen that the expression ( 1 + 2 + 3 + 4 + 5 ) ( 2 + 3 + 4 + 5 + 6 ) ( 3 + 4 + 5 + 6 + 7 ) gives the sum of all possible ways that 3 numbers from the given ranges can be multiplied together. The total number of ways a , b , and c can be combined is 5 ⋅ 5 ⋅ 5 = 1 2 5 , so the mean average of the values given by a b c is equal to: = = 1 2 5 1 ( 1 + 2 + 3 + 4 + 5 ) ( 2 + 3 + 4 + 5 + 6 ) ( 3 + 4 + 5 + 6 + 7 ) 1 2 5 1 ( 1 5 ) ( 2 0 ) ( 2 5 ) 6 0