The total number of 4-digit numbers which can | vedclass

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

Question

(B) We need to find the number of permutations of ${1, 2, 3, 4}$ such that the patterns $12, 23, 34$ do not occur.Total permutations of $4$ digits is

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(B) We need to find the number of permutations of ${1, 2, 3, 4}$ such that the patterns $12, 23, 34$ do not occur.Total permutations of $4$ digits is $4! = 24$.Let $S$ be the set of all permutations. Let $A_1$ be the set of permutations containing $12$,$A_2$ be the set containing $23$,and $A_3$ be the set containing $34$.We want to find $|S| - |A_1 \cup A_2 \cup A_3|$.Using the Principle of Inclusion-Exclusion:$|A_1| = |A_2| = |A_3| = 3! = 6$.$|A_1 \cap A_2|$ (contains $123$) = $2! = 2$.$|A_2 \cap A_3|$ (contains $234$) = $2! = 2$.$|A_1 \cap A_3|$ (contains $12$ and $34$) = $2! = 2$.$|A_1 \cap A_2 \cap A_3|$ (contains $1234$) = $1! = 1$.$|A_1 \cup A_2 \cup A_3| = (6+6+6) - (2+2+2) + 1 = 18 - 6 + 1 = 13$.Number of valid permutations = $24 - 13 = 11$.

The total number of $4$-digit numbers which can be formed using the digits $1, 2, 3, 4$ without repetition such that the digit $n+1$ never immediately follows the digit $n$ (for $n=1, 2, 3$) is:

Similar Questions

How many numbers lying between $999$ and $10000$ can be formed with the help of the digits $0, 2, 3, 6, 7, 8$ when the digits are not to be repeated?

The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to

How many $4$-digit numbers are there with distinct digits?

If $^{56}P_{r+6} : ^{54}P_{r+3} = 30800 : 1$,then $r = $

There are $4$ notes of Rs. $100$ and $5$ other notes of denominations Rs. $1$,Rs. $2$,Rs. $5$,Rs. $20$,and Rs. $50$. These notes are to be distributed among $3$ children such that each child receives at least one note of Rs. $100$. Find the total number of ways of distribution.

Vedclass Test Series

Exam Paper Generator

Online Exam Module