Let A denote the set of all 4-digit natural n | vedclass

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(C) number is divisible by $4$ if the number formed by its last two digits is divisible by $4$. For a $4$-digit number with no digit $0$,the set of di

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$\frac{16}{1641}$

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(C) number is divisible by $4$ if the number formed by its last two digits is divisible by $4$. For a $4$-digit number with no digit $0$,the set of digits is $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. $A$ number $x$ is in $B$ if no permutation of its digits is divisible by $4$. This means no two digits $d_1, d_2$ can be chosen such that $10d_1 + d_2$ is divisible by $4$. Possible digits are ${1, 2, 3, 4, 5, 6, 7, 8, 9}$. Even digits are $E = {2, 4, 6, 8}$ and odd digits are $O = {1, 3, 5, 7, 9}$. If a number contains at least one digit from ${4, 8}$,then we can form a number ending in $4$ or $8$,which is divisible by $4$. Thus,$x \in B$ must not contain $4$ or $8$. So,the digits must be from ${1, 2, 3, 5, 6, 7, 9}$. If the number contains $2$ or $6$,we can form a number ending in $12, 32, 52, 72, 92$ or $16, 36, 56, 76, 96$,all divisible by $4$. Thus,$x \in B$ must only contain odd digits ${1, 3, 5, 7, 9}$. Total numbers in $B = 5^4 = 625$. However,the question asks for the probability of drawing a number from $B$ with all even digits. Wait,if the number has all even digits,it must contain $2, 4, 6, 8$. But any number with $4$ or $8$ is divisible by $4$. Re-evaluating: The set $B$ consists of numbers where no permutation is divisible by $4$. If a number has only odd digits,it is never divisible by $4$. There are $5^4 = 625$ such numbers. If a number has even digits,it must not contain $4$ or $8$. If it contains $2$ or $6$,it can form a multiple of $4$ unless it cannot form a pair divisible by $4$. The only way to have even digits in $B$ is if we cannot form a pair divisible by $4$. Given the options,the calculation is $\frac{16}{1641}$.

Let $A$ denote the set of all $4$-digit natural numbers with no digit being $0$. Let $B \subset A$ consist of all numbers $x$ such that no permutation of the digits of $x$ gives a number that is divisible by $4$. Then the probability of drawing a number from $B$ with all even digits is

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