JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 23

Divisibility by 11. The digits sum to 31. With 4 odd-place and 3 even-place digits, $S_{\text{odd}}-S_{\text{even}}$ must be a multiple of 11, so it is $\pm11$: the even places sum to 10 or 21.

Choices for the even places. Sum 10: $\{1,2,7\},\{1,4,5\},\{2,3,5\}$. Sum 21: $\{5,7,9\}$. Four sets.

Arrange. Each gives $3!\times4!=144$ numbers, so $4\times144=576$.

Answer: 576