GATE DA 2026 Q20: Probability all parts are positive in n1+n2+n3+n4 = 20

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation $n_1 + n_2 + n_3 + n_4 = 20$ .

Which of the following is the probability that all of $n_1, n_2, n_3, n_4$ in the provided solution are positive?

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Answer: A (official GATE 2026 final answer key)

Concept used: Stars and bars: non-negative solutions $\binom{n+k-1}{k-1}$ , positive solutions $\binom{n-1}{k-1}$ .

First step: With $n = 20$ and $k = 4$ : $\binom{23}{3}$ solutions in all, $\binom{19}{3}$ with every part positive.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key