Number of ways of selecting three integers from \( \{1,2,3 \), \( \ldots, n\} \) if their sum is... VIDEO
Number of ways of selecting three integers from \( \{1,2,3 \), \( \ldots, n\} \) if their sum is divisible by 3 is
(1) \( 3\left({ }^{n / 3} C_{3}\right)+(n / 3)^{3} \) if \( n=3 k, k \in N \)
(2) \( 2\left({ }^{(n-1) / 3} C_{3}\right)+\left({ }^{(n+2) / 3} C_{3}\right)+((n-1) / 3)^{2}((n+2) / 3) \) if \( n=3 k+1, k \in N \)
(3) \( 2\left({ }^{(n-1) / 3} C_{3}\right)+\left({ }^{(n+2) / 3} C_{3}\right)+((n-1) / 3)^{2}((n+2) / 3) \) if \( n=3 k+2, k \in N \)
(4) independent of \( n \)
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