Exercise 1718Combinatorics › Inclusion-exclusionOlympiadAdded 3 Nov 2026

Path counting via IE

#IE#paths#obstacles

Statement

Count monotone paths from $(0,0)$ to $(6,4)$ that avoid the point $(3,2)$ .

Hint

Total minus paths through $(3,2)$ .

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Total. $\binom{10}{6}=210$ .

Through $(3,2)$ . Paths $(0,0)\to(3,2)$ times paths $(3,2)\to(6,4)$ : $\binom{5}{3}\cdot\binom{5}{3}=10\cdot10=100$ .

Avoid. $210-100=110$ .

Conclusion. $110$ paths avoid $(3,2)$ .