Find the number of different 8-letter arrange | vedclass

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

Question

(A) The word $DAUGHTER$ consists of $8$ distinct letters: $D, A, U, G, H, T, E, R$.There are $3$ vowels: $A, U, E$ and $5$ consonants: $D, G, H, T, R$

Vedclass · Accepted Answer

$4320$

Vedclass · Answer

(A) The word $DAUGHTER$ consists of $8$ distinct letters: $D, A, U, G, H, T, E, R$.There are $3$ vowels: $A, U, E$ and $5$ consonants: $D, G, H, T, R$.Since the vowels must occur together,we treat them as a single unit or object: $(AUE)$.Now,we have $5$ consonants plus $1$ unit of vowels,making a total of $6$ objects.These $6$ objects can be arranged in $6!$ ways.Within the vowel unit,the $3$ vowels $(A, U, E)$ can be arranged among themselves in $3!$ ways.By the multiplication principle,the total number of arrangements is $6! 	imes 3!$.$6! = 720$ and $3! = 6$.Total arrangements $= 720 	imes 6 = 4320$.

Find the number of different $8$-letter arrangements that can be made from the letters of the word $DAUGHTER$ such that all vowels occur together.

Similar Questions

How many arrangements can be made using the letters of the word $CALCUTTA$?

In a train,five seats are vacant. In how many ways can three passengers sit?

The number of ways in which $3$ men and $3$ women can be arranged in a row of $6$ seats,such that the first and last seats must be filled by men is

How many numbers between $999$ and $10,000$ can be formed using the digits $0, 2, 3, 6, 7, 8$ without repetition?

If $^n{P_r} = 720 \times {^n{C_r}}$,then $r$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module