Among the 4-digit numbers that can be formed | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) number is divisible by $6$ if it is divisible by both $2$ and $3$.For a number to be divisible by $2$,the units digit must be $2, 4,$ or $6$.For a

Vedclass · Accepted Answer

$60$

Vedclass · Answer

(A) number is divisible by $6$ if it is divisible by both $2$ and $3$.For a number to be divisible by $2$,the units digit must be $2, 4,$ or $6$.For a number to be divisible by $3$,the sum of its digits must be divisible by $3$.Let the $4$ digits be $d_1, d_2, d_3, d_4$ where $d_4 \in \{2, 4, 6\}$.Case $I$: Units digit is $2$. The sum of the remaining $3$ digits must be $3k - 2$. Possible sets of $3$ digits from $\{1, 3, 4, 5, 6\}$ are $\{1, 4, 6\}, \{1, 5, 6\}, \{3, 4, 5\}$. Each set can be arranged in $3! = 6$ ways. Total $= 3 	imes 6 = 18$.Case $II$: Units digit is $4$. The sum of the remaining $3$ digits must be $3k - 4$. Possible sets of $3$ digits from $\{1, 2, 3, 5, 6\}$ are $\{1, 2, 6\}, \{1, 5, 6\}, \{2, 3, 6\}$. Each set can be arranged in $3! = 6$ ways. Total $= 3 	imes 6 = 18$.Case $III$: Units digit is $6$. The sum of the remaining $3$ digits must be $3k - 6$. Possible sets of $3$ digits from $\{1, 2, 3, 4, 5\}$ are $\{1, 2, 3\}, \{1, 3, 5\}, \{2, 3, 4\}, \{3, 4, 5\}$. Each set can be arranged in $3! = 6$ ways. Total $= 4 	imes 6 = 24$.Total numbers $= 18 + 18 + 24 = 60$.

Among the $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6$ without repeating any digit,the number of numbers which are divisible by $6$ is:

Similar Questions

If ${}^n C_{r-1}=36$, ${}^n C_r=84$, and ${}^n C_{r+1}=126$, then the value of ${}^n C_8$ is

In how many ways can $3n$ distinct objects be distributed equally among $3$ persons?

The number of natural numbers,between $212$ and $999$,such that the sum of their digits is $15$,is . . . . . .

In how many ways can $5$ distinct balls be distributed among $3$ persons such that each person receives at least one ball?

$A$ number is called a palindrome if it reads the same backward as well as forward. For example,$285582$ is a six-digit palindrome. The number of six-digit palindromes,which are divisible by $55$,is ...... .

Vedclass Test Series

Exam Paper Generator

Online Exam Module