The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

Question

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

Smart Achievers · Accepted Answer

(81)

Total number of 6-digit numbers formed by the digits 4, 5, 9 and divisible by 6 is when all the digits are same.

$444444 \to \frac{6!}{6!} = 1$

$Taking two digits (4, 5), 555444 \to \frac{5!}{3! 2!} = 10$

$For (4, 9), 999444 \to \frac{5!}{3! 2!} = 10$

$Taking three digits: 459444 \to \frac{5!}{3!} = 20$

$559554 \to \frac{5!}{4!} = 5, 959994 \to \frac{5!}{4!} = 5$

$959454 \to \frac{5!}{2! 2!} = 30$

$Total number = 20 + 5 + 5 + 30 + 10 + 10 + 1 = 81$

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