Difference between revisions of "2020 AMC 10A Problems/Problem 18"
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- I believe this is incorrect. When finding the evens, <math>2 \cdot 4</math> is incorrect because you are assuming the first number is always even/odd. <math>(C)</math> is the correct answer. | - I believe this is incorrect. When finding the evens, <math>2 \cdot 4</math> is incorrect because you are assuming the first number is always even/odd. <math>(C)</math> is the correct answer. | ||
− | == Solution == | + | == Solution 2 == |
Consider parity. We need exactly one term to be odd, one term to be even. Because of symmetry, we can set <math>ad</math> to be odd and <math>bc</math> to be even, then multiply by <math>2.</math> If <math>ad</math> is odd, both <math>a</math> and <math>d</math> must be odd, therefore there are <math>2\cdot2=4</math> possibilities for <math>ad.</math> Consider <math>bc.</math> Let us say that <math>b</math> is even. Then there are <math>2\cdot4=8</math> possibilities for <math>bc.</math> However, <math>b</math> can be odd, in which case we have <math>2\cdot2=4</math> more possibilities for <math>bc.</math> Thus there are <math>12</math> ways for us to choose <math>bc</math> and <math>4</math> ways for us to choose <math>ad.</math> Therefore, also considering symmetry, we have <math>2*4*12=96</math> total values of <math>ad-bc.</math> <math>(C)</math> | Consider parity. We need exactly one term to be odd, one term to be even. Because of symmetry, we can set <math>ad</math> to be odd and <math>bc</math> to be even, then multiply by <math>2.</math> If <math>ad</math> is odd, both <math>a</math> and <math>d</math> must be odd, therefore there are <math>2\cdot2=4</math> possibilities for <math>ad.</math> Consider <math>bc.</math> Let us say that <math>b</math> is even. Then there are <math>2\cdot4=8</math> possibilities for <math>bc.</math> However, <math>b</math> can be odd, in which case we have <math>2\cdot2=4</math> more possibilities for <math>bc.</math> Thus there are <math>12</math> ways for us to choose <math>bc</math> and <math>4</math> ways for us to choose <math>ad.</math> Therefore, also considering symmetry, we have <math>2*4*12=96</math> total values of <math>ad-bc.</math> <math>(C)</math> | ||
Revision as of 23:26, 31 January 2020
Contents
[hide]Problem
Let be an ordered quadruple of not necessarily distinct integers, each one of them in the set For how many such quadruples is it true that is odd? (For example, is one such quadruple, because is odd.)
Solution 1
In order for to be odd, consider parity. We must have (even)-(odd) or (odd)-(even). There are ways to pick numbers to obtain a even product. There are ways to obtain an odd product. Therefore, the total amount of ways to make odd is .
-Midnight
- I believe this is incorrect. When finding the evens, is incorrect because you are assuming the first number is always even/odd. is the correct answer.
Solution 2
Consider parity. We need exactly one term to be odd, one term to be even. Because of symmetry, we can set to be odd and to be even, then multiply by If is odd, both and must be odd, therefore there are possibilities for Consider Let us say that is even. Then there are possibilities for However, can be odd, in which case we have more possibilities for Thus there are ways for us to choose and ways for us to choose Therefore, also considering symmetry, we have total values of
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 10 Problems and Solutions |
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