
Find the number of different words which can be formed from the letters of the word \[LUCKNOW\] when the vowels always occupy even places.
A. \[120\]
B. \[720\]
C. \[400\]
D. None of these
Answer
161.4k+ views
Hint: First, find the number of consonants and vowels present in the given word \[LUCKNOW\]. Then, arrange the letters in even and odd places. In the end, find how many ways the letters can be arranged to get the required answer.
Formula Used: \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]
Complete step by step solution: The given word is \[LUCKNOW\].
Total number of letters present in the word: 7
Number of consonants: L, C, K, N, and W: 5
Number of vowels: U, and O: 2
Since, in the given word 2 vowels O, and U are present. There are 3 even places.
So, the number of ways of arranging 2 vowels at 3 places are: \[{}^3{P_2}\]
Also, there are 5 consonants present in the word \[LUCKNOW\].
So, the number of ways of arranging 5 consonants at 4 places are: \[{}^5{P_4}\]
Therefore, the number of words formed in which vowels occupy the even places are:
\[{}^3{P_2} \times {}^5{P_4} = \dfrac{{3!}}{{\left( {3 - 2} \right)!}} \times \dfrac{{5!}}{{\left( {5 - 4} \right)!}}\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = \dfrac{{3!}}{{1!}} \times \dfrac{{5!}}{{1!}}\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = 6 \times 120\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = 720\]
Option ‘B’ is correct
Note: Permutation shows the number of possible arrangements of the objects when the order of the arrangement of the objects matters.
While solving this type of problem use the permutation method.
Formula Used: \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]
Complete step by step solution: The given word is \[LUCKNOW\].
Total number of letters present in the word: 7
Number of consonants: L, C, K, N, and W: 5
Number of vowels: U, and O: 2
Since, in the given word 2 vowels O, and U are present. There are 3 even places.
So, the number of ways of arranging 2 vowels at 3 places are: \[{}^3{P_2}\]
Also, there are 5 consonants present in the word \[LUCKNOW\].
So, the number of ways of arranging 5 consonants at 4 places are: \[{}^5{P_4}\]
Therefore, the number of words formed in which vowels occupy the even places are:
\[{}^3{P_2} \times {}^5{P_4} = \dfrac{{3!}}{{\left( {3 - 2} \right)!}} \times \dfrac{{5!}}{{\left( {5 - 4} \right)!}}\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = \dfrac{{3!}}{{1!}} \times \dfrac{{5!}}{{1!}}\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = 6 \times 120\]
\[ \Rightarrow {}^3{P_2} \times {}^5{P_4} = 720\]
Option ‘B’ is correct
Note: Permutation shows the number of possible arrangements of the objects when the order of the arrangement of the objects matters.
While solving this type of problem use the permutation method.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
