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Derek (no school given) offered the following insights into this problem: - and nicely explained too - thank you Derek.
n-row pyramid | Ways to count 1-2-3-... |
1 | 1 |
2 | 3 |
3 | 7 |
4 | 15 |
... | ... |
Editor's comment: at this point it may be worth mentioniing that the sum of the binomial coefficients are powers of 2. Why?
Chen of the Chinese High School took a slightly different approach (similar to the one offered by Andrei of Tudor Vianu National College):
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?