Difference between revisions of "2007 AMC 12A Problems/Problem 12"
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==Problem== | ==Problem== | ||
| − | Integers a, b, c, and d, not necessarily distinct, are chosen | + | Integers <math>a, b, c,</math> and <math>d</math>, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the [[probability]] that <math>ad-bc</math> is [[even]]? |
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| + | <math>\mathrm{(A)}\ \frac 38\qquad \mathrm{(B)}\ \frac 7{16}\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 9{16}\qquad \mathrm{(E)}\ \frac 58</math> | ||
==Solution== | ==Solution== | ||
| + | The only times when <math>ad-bc</math> is even is when <math>ad</math> and <math>bc</math> are of the same [[parity]]. The chance of <math>ad</math> being odd is <math>\frac 12 \cdot \frac 12 = \frac 14</math>, so it has a <math>\frac 34</math> probability of being even. Therefore, the probability that <math>ad-bc</math> will be even is <math>\displaystyle \left(\frac 14\right)^2+\left(\frac 34\right)^2=\frac 58\ \mathrm{(E)}</math>. | ||
| − | + | ==See also== | |
| + | {{AMC12 box|year=2007|ab=A|num-b=11|num-a=13}} | ||
| − | + | [[Category:Introductory Combinatorics Problems]] | |
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Revision as of 14:33, 10 September 2007
Problem
Integers 
and 
, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that 
is even?
Solution
The only times when is even is when 
and 
are of the same parity. The chance of being odd is 
, so it has a 
probability of being even. Therefore, the probability that will be even is 
.
See also
| 2007 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 | |
