In the plane, the points with integer coordinates are called lattice points. Suppose a flea in the plane jumps from one lattice point to another. Each jump is one unit to the right, one unit to the left, one unit up, or one unit down. If the flea starts at the origin and makes exactly 10 jumps, how many lattice points could possibly be the final landing place of the flea?

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To figure out how many lattice points could be the final landing place of the flea, we can consider the possible combinations of jumps the flea can make.

Since each jump can be one unit to the right, one unit to the left, one unit up, or one unit down, we have four possible directions for each jump. With 10 jumps in total, there are 4^10 (four to the power of 10) possible combinations of jumps.

However, not all these combinations will result in distinct final landing points. To determine the distinct landing points, we can consider the net displacement of the flea after its 10 jumps.

The flea starts at the origin, which has coordinates (0, 0). Each time it jumps one unit to the right, its x-coordinate increases by 1. Each time it jumps one unit to the left, its x-coordinate decreases by 1. Similarly, each upward jump increases the y-coordinate by 1, and each downward jump decreases the y-coordinate by 1.

So, after 10 jumps, the x-coordinate of the flea can be anything from -10 to +10, and the y-coordinate can also be anything from -10 to +10.

Therefore, there are a total of (10+10+1) * (10+10+1) = 21 * 21 = 441 possible lattice points that could be the final landing place of the flea.

Hence, the flea could possibly end up on 441 different lattice points.