The number of arrangements of all digits of 1 | vedclass

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

Question

(B) The total number of arrangements of $5$ distinct digits is $5! = 120$.We need to find the number of arrangements where at least $3$ digits do not

Vedclass · Accepted Answer

$109$

Vedclass · Answer

(B) The total number of arrangements of $5$ distinct digits is $5! = 120$.We need to find the number of arrangements where at least $3$ digits do not appear in their original positions.Let $S$ be the set of all permutations. We use the principle of derangements $(D_n)$.The number of ways to choose $k$ digits that are $NOT$ in their original positions is given by $\binom{5}{k} 	imes D_k$,where $D_k$ is the number of derangements of $k$ objects.For $k=3$: $\binom{5}{3} 	imes D_3 = 10 	imes 2 = 20$.For $k=4$: $\binom{5}{4} 	imes D_4 = 5 	imes 9 = 45$.For $k=5$: $\binom{5}{5} 	imes D_5 = 1 	imes 44 = 44$.Total arrangements = $20 + 45 + 44 = 109$.

The number of arrangements of all digits of $12345$ such that at least $3$ digits will not come in their original positions is

Similar Questions

There are $n$ white and $n$ black balls marked $1, 2, 3, ..., n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours are

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$,$Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee must have $2$ trainees and $3$ research associates?

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

If ${}^n{P_4} : {}^n{P_5} = 1 : 2$,then $n = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module