<p>here are <img alt="$8!= 40320$" height="14" src="https://latex.artofproblemsolving.com/d/3/1/d310bd62c7ab6a1cb67cf2c1a65a175edb639806.png" width="86" /> eight-digit positive integers that use each of the digits <img alt="$1, 2, 3, 4, 5, 6, 7, 8$" height="17" src="https://latex.artofproblemsolving.com/5/0/9/509874be825c3239ee5d4b41924c90e4c7d652c0.png" width="131" /> exactly once. Let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> be the number of these integers that are divisible by <img alt="$22$" height="12" src="https://latex.artofproblemsolving.com/1/d/1/1d111a5e00ee5ea6700bb629994dc629874c505b.png" width="17" />. Find the difference between <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> and <img alt="$2025$" height="14" src="https://latex.artofproblemsolving.com/7/8/d/78db23e86da8fe319c63bdb5eaf6d3fd8fefa17b.png" width="38" />.</p>

Question No: 403

here are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, 7, 8$ exactly once. Let $N$ be the number of these integers that are divisible by $22$ . Find the difference between $N$ and $2025$ .

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Tags: Combinatorics Number Theory

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