Unchanged after ten swings

EasyProbability Brainteasers~4m

An asset moves up $20\%$ or down $20\%$ each day (from the previous day's price), each with probability $\tfrac12$ . What is the probability that after $10$ days its price is exactly back to the starting value?

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After $u$ ups and $d$ downs the factor is $(6/5)^u(4/5)^d$ . Set it to $1$ .
That needs $6^u4^d = 5^{10}$ - compare prime factorizations.

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Asked at: Jane Street, Optiver

Unchanged after ten swings

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