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# counting pairs

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How many ordered pairs \$(x, y)\$ satisfy BOTH conditions below?

\$\bullet\$  Condition I: \$x = 1\$ or \$y = 0\$ or \$y = 2\$ or \$x = 0\$ or \$x = -1\$

\$\bullet\$  Condition II: \$x = 0\$ or \$x = 2\$ or \$y = 1\$ or \$y = 0\$ or \$y = -1\$

Aug 16, 2023

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To find the ordered pairs \((x, y)\) that satisfy both conditions, we need to identify the values that satisfy both Condition I and Condition II.

For Condition I, the possible pairs are:

1. \(x = 1\) and \(y\) can be any value.
2. \(y = 0\) and \(x\) can be any value.
3. \(y = 1\) and \(x\) can be any value.
4. \(x = 0\) and \(y\) can be any value.
5. \(x = -1\) and \(y\) can be any value.

For Condition II, the possible pairs are:

1. \(x = 0\) and \(y\) can be any value.
2. \(x = 2\) and \(y\) can be any value.
3. \(y = 1\) and \(x\) can be any value.
4. \(y = 0\) and \(x\) can be any value.
5. \(y = -1\) and \(x\) can be any value.

To satisfy both conditions, the ordered pairs must match the common values in both lists. The common values are \(x = 0\) and \(y = 0\) or \(y = 1\).

Therefore, the ordered pairs that satisfy both conditions are \((0, 0)\) and \((0,1)\), and there are \(\boxed{2}\) such pairs.

Aug 16, 2023