If a number is chosen at random from the four | vedclass

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

Question

(B) The total number of four-digit numbers formed using the digits $\{0, 1, 2, 3, 4, 6\}$ without repetition is calculated as follows: The first digit

Vedclass · Accepted Answer

$\frac{17}{50}$

Vedclass · Answer

(B) The total number of four-digit numbers formed using the digits $\{0, 1, 2, 3, 4, 6\}$ without repetition is calculated as follows: The first digit cannot be $0$,so there are $5$ choices. The remaining three positions can be filled in $5 	imes 4 	imes 3 = 60$ ways. Thus,total numbers $= 5 	imes 60 = 300$.$A$ number is divisible by $4$ if its last two digits form a number divisible by $4$. The possible pairs for the last two digits are $\{04, 12, 16, 20, 24, 32, 36, 40, 60, 64\}$.Case $1$: Pairs containing $0$ $(\{04, 20, 40, 60\})$: There are $4$ such pairs. For each,the remaining $2$ positions can be filled by the remaining $4$ digits in $4 	imes 3 = 12$ ways. Total $= 4 	imes 12 = 48$.Case $2$: Pairs not containing $0$ $(\{12, 16, 24, 32, 36, 64\})$: There are $6$ such pairs. For each,the first digit cannot be $0$ or the two digits already used,leaving $3$ choices for the first digit and $3$ choices for the second digit. Total $= 6 	imes (3 	imes 3) = 6 	imes 9 = 54$.Total favourable outcomes $= 48 + 54 = 102$.Required probability $= \frac{102}{300} = \frac{17}{50}$.

If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is

Similar Questions

If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$,find $x$.

The sum of all the $4$-digit numbers formed by taking all the digits from ${0, 3, 6, 9}$ without repetition is

Find the remainder when $(1!)^2 + (2!)^2 + (3!)^2 + \dots + (100!)^2$ is divided by $10^2$.

The sum of all $4$-digit numbers that can be formed using the digits $2, 3, 4, 5, 6$ without repetition is:

For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

Vedclass Test Series

Exam Paper Generator

Online Exam Module