The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is .

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🧠 Reading the Question — Partition Digits Into Two Sum-Equal Groups

We form -digit numbers using all of  (each exactly once), divisible by .

The digit sum is

Divisibility-by- rule: if  is the sum of digits at odd positions and  at even positions, then .

A -digit number has  odd positions () and  even positions (). So  is the sum of  digits from our set, and  the sum of the remaining .

With : , so . Since  is a sum of  digits from , its range is . Values of  congruent to  mod  in this rang

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