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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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Answer» Digits are 1, 2, 3, 4, 5, 7, 9 Multiple of 11 → Difference of sum at even & odd place is divisible by 11. Let number of the form abcdefg \(\therefore\) (a + c + e + g) – (b + d + f) = 11x a + b + c + d + e + f = 31 \(\therefore\) either a + c + e + g = 21 or 10 \(\therefore\) b + d + f = 10 or 21 Case- 1 a + c + e + g = 21 b + d + f = 10 (b, d, f) ∈ {(1, 2, 7) (2, 3, 5) (1, 4, 5)} (a, c, e, g) ∈ {(1, 4, 7, 9), (3, 4, 5, 9), (2, 3, 7, 9)} \(\therefore\) Total number in case-1 = (3! × 3) (4!) = 432 Case- 2 a + c + e + g = 10 b + d + f = 21 (a, b, e, g) ∈ {1, 2, 3, 4)} (b, d, f) & {(5, 7, 9)} \(\therefore\) Total number in case 2 = 3! × 4! = 144 \(\therefore\) Total numbers = 144 + 432 = 576 |
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