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Challenge 256: Paula's Triangle

Paula is drawing number triangles. On the bottom row of each triangle, there are 5 numbers. The second row has 4 numbers, and so on, until the fifth row has a single number. She always fills the bottom row with the numbers 1 to 5 in some order.

(a) She then fills the rest of the triangle so that every number is the sum of the two numbers below it. What is the largest possible number she can get at the top of the triangle? If she places the numbers 1-5 on the bottom row at random, what is the probability she will obtain this maximum?

(b) Trying something different, she fills in the rest of the triangle so that every number is the difference of the two numbers below it, and all numbers in her triangle are non-negative. What is the largest possible number she can get at the top of the triangle? If she places the numbers 1-5 on the bottom row at random, what is the probability she will obtain this maximum?

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