An ant has to walk from the left most lower corner to top most upper corner of $4 \times 4$ square. However, it can take a right step or an upward step, but not necessarily in order. Calculate the number of paths.
2026-03-31 06:17:00.1774937820
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An ant is to walk from $A$ to $B$. Calculate the number of paths.
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Realise that the ant in question must take 4 steps to the right and 4 upwards. Representing a step towards right as R and an upward step as U, the ant can choose paths like RRRUUUUR, URURURUR, etc.
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Whatever may be path chosen the ant has to take 4 upward steps and 4 righgward steps.
Therefore there are $\frac{8!}{4!4!}$ Paths...i,e $70$ paths
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The answer to your question are staircase walks that are related to Dyck paths.
In general there are $\binom{m+n}{n}$ such paths. In your case $n=m=4$.
HINT: In total, there are $4$ upward moves ($U$) and $4$ rightward moves ($R$). So for example $UUUURRRR$ is one of the ways ant can use. So the problem can be restated as "How many different words (meaningful or not meaningful) can be formed with $4\ U$'s and $4\ R$'s?".