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Question
How many different 6-digit numbers can be formed using digits in the number 659942? How many of them are divisible by 4?
Solution
A 6-digit number is to be formed using digits of 659942, in which 9 repeats twice.
∴ Total number of arrangements
= `(6!)/(2!)`
= `(6 xx 5 xx 4 xx 3 xx 2!)/(2!)`
= 360
∴ 360 different 6-digit numbers can be formed.
For a number to be divisible by 4, the last two digits should be divisible by 4 i.e. 24, 52, 56, 64, 92, or 96.
Case I: When the last two digits are 24, 52, 56 or 64.
As the digit 9 repeats twice in the remaining four numbers, the number of arrangements
= `(4!)/(2!)`
= 12
∴ 6-digit numbers that are divisible by 4 so formed are 12 + 12 + 12 + 12 = 48.
Case II: When the last two digits are 92 or 96.
As each of the remaining four numbers is distinct, the number of arrangements = 4! = 24
∴ 6-digit numbers that are divisible by 4 so formed are 24 + 24 = 48.
∴ Total number of arrangements from both these cases is 48 + 48 = 96.
Thus, 96 6-digit numbers can be formed that are divisible by 4.
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