<p>Find the number of permutations <img alt="$x_1, x_2, x_3, x_4, x_5$" height="11" src="https://latex.artofproblemsolving.com/1/7/3/173d64f6629624d194f54d23810583b4162529c7.png" width="120" /> of numbers <img alt="$1, 2, 3, 4, 5$" height="16" src="https://latex.artofproblemsolving.com/9/f/a/9faa9b63684d102b90b612f4f964224e5437ce30.png" width="77" /> such that the sum of five products<img alt="\[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\]" height="13" src="https://latex.artofproblemsolving.com/5/f/8/5f85b1e5c0060aca6f470f54f8117ac7500be017.png" width="355" />is divisible by <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" />.</p>

Question No: 360

Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$ is divisible by $3$ .

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Tags: Algebra Combinatorics

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