Question:
How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?
How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?
Show Hint
For forming words using selected vowels and consonants, first choose the required letters and then arrange all selected letters using factorial.
Updated On: Jun 25, 2026
- \(120\)
- \(21600\)
- \(720\)
- \(3600\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Identify vowels and consonants in DAUGHTER.
The word is DAUGHTER The vowels are \[ A,U,E \] So, the number of vowels is \[ 3 \] The consonants are \[ D,G,H,T,R \] So, the number of consonants is \[ 5 \]
Step 2: Select 2 vowels.
Number of ways of selecting \(2\) vowels from \(3\) vowels is \[ {}^3C_2=3 \]
Step 3: Select 3 consonants.
Number of ways of selecting \(3\) consonants from \(5\) consonants is \[ {}^5C_3=10 \]
Step 4: Arrange the selected letters.
Total selected letters are \[ 2+3=5 \] These \(5\) letters can be arranged in \[ 5! \] ways. Now, \[ 5!=120 \]
Step 5: Find the total number of words.
Total number of words is \[ {}^3C_2\times {}^5C_3\times 5! \] \[ =3\times 10\times 120 \] \[ =3600 \]
Step 6: Final conclusion.
Therefore, \[ \boxed{3600} \]
The word is DAUGHTER The vowels are \[ A,U,E \] So, the number of vowels is \[ 3 \] The consonants are \[ D,G,H,T,R \] So, the number of consonants is \[ 5 \]
Step 2: Select 2 vowels.
Number of ways of selecting \(2\) vowels from \(3\) vowels is \[ {}^3C_2=3 \]
Step 3: Select 3 consonants.
Number of ways of selecting \(3\) consonants from \(5\) consonants is \[ {}^5C_3=10 \]
Step 4: Arrange the selected letters.
Total selected letters are \[ 2+3=5 \] These \(5\) letters can be arranged in \[ 5! \] ways. Now, \[ 5!=120 \]
Step 5: Find the total number of words.
Total number of words is \[ {}^3C_2\times {}^5C_3\times 5! \] \[ =3\times 10\times 120 \] \[ =3600 \]
Step 6: Final conclusion.
Therefore, \[ \boxed{3600} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Permutations Questions
- All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order. Then the rank of the word TETLER is:
- 5-digit numbers are formed by using the digits 0, 1, 2, 3, 5, 7 without repetition and all of them are arranged in ascending order. Then the rank of the number 70513 is:
- The number of positive integers less than 10000 which contain the digit 5 at least once is
- 5 men and 4 women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $ \alpha $, and the number of arrangements in which they are not together is $ \beta $, then $ \frac{\alpha}{\beta} = $
- From the word "CURVE", how many 3-letter words can be formed out of all 2-letter or more combinations (with all distinct letters)? Find probability of getting a 3-letter word.
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions