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In how many ways can one arrange five 1’s and five -1’s so that all ten partial sums (starting with the first summand) are nonnegative?
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When n=4 then we can draw like this,

Number of Leaves = Total Arrangement = 5

If you draw a Tree like this for n=5, then you will get 14.

I am not sure but this looks like Catalan Number i.e. if $n=4$ then Catalan Number $C_3=5$ and for $n=5$ $C_4=14$

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The answer is 42 which maps to $C_{5}$. But why $C_{5}$?