Question

Both of the correct answers equal 2,400 points. Why do they have the same value?

Answer 1

I don't have the original problem or the two expressions you used, so I can't show the specific algebraic identity that makes them equal. Could you paste the two answers or the work you used?

Meanwhile, here are the typical reasons two different-looking answers turn out to be the same number (and how to check which one applies):

1. Algebraic identity / simplification
- The two expressions reduce to the same algebraic form after expanding/factoring.
- How to check: subtract one from the other and simplify; if the difference is 0, they are equal.

2. Equivalent factorings / rearrangement
- The same factors are just arranged differently (e.g., 48·50 and 60·40 both equal 2400 because 48·50 = (24·2)·50 = 24·100 = 2400 and 60·40 = 6·10·40 = 2400).
- How to check: factor both sides and compare.

3. Two counting methods of the same set (bijective argument)
- You counted the same objects in two different ways (e.g., count by rows vs by columns), so both counts must match.
- How to check: describe a one-to-one correspondence between the items counted in each method.

4. Symmetry or invariance in the problem
- A symmetry (geometric or combinatorial) forces different-looking quantities to be equal.
- How to check: point out the symmetry and show it maps one quantity to the other.

5. Use of a standard identity (binomial coefficients, sums, etc.)
- Two combinatorial expressions may be equal due to an identity (like Pascal's identities, sum formulas, etc.).
- How to check: apply the known identity or prove it algebraically.

If you paste the two expressions or the original problem, I’ll show exactly why they both equal 2400 (either by algebraic simplification or by giving a counting/bijection argument).