How many groups each of 2 vowels and 3 consonants can be formed from the letter of the word computer?

Last updated at Jan. 13, 2022 by Teachoo

This video is only available for Teachoo black users

Misc 1 How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER? Number ways of selecting 2 vowels & 3 consonants = 3C2 × 5C3 = 3!/2!(3 − 2)! × 5!/3!(5 − 3)! = 3!/2!1! × 5!/3!2! = 30 Now, Each of these 5 letters can be arranged in 5 ways Number of arrangements = 5P5 = 5!/(5 − 5)! = 5!/0! = 5! = 5 × 4 × 3 × 2 × 1 = 120 Thus, Total number of words = Number of ways of selecting × Number of arrangements = 30 × 120 = 3600

Answer

Verified

Hint: Count the number of vowels and consonants in the word DAUGHTER. Let the counts be x, y respectively. The required words should have 2 vowels and 3 consonants in it. So the no. of words that contains 2 vowels and 3 consonants which can be formed from the letters of DAUGHTER is ${}^x{C_2} \times {}^y{C_3}$

Complete step-by-step answer:

We are given to find the number of words that can be formed from the letters of the word DAUGHTER which contains 2 vowels and 3 consonants.The given word is DAUGHTER. This word has 3 vowels, A, U, E, and 5 consonants, D, G, H, T and R.So the required words should have 2 vowels from A, U and E; 3 consonants from D, G, H, T and R.And the order of the letters is not specific, which means the letters can be used in any order. So we have to use combinations. So the no. of words will be ${}^3{C_2} \times {}^5{C_3}$, selecting any 2 from 3 vowels and selecting any 3 from 5 consonants.$ {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} \\ {}^3{C_2};n = 3,c = 2 \\ {}^3{C_2} = \dfrac{{3!}}{{2!\left( {3 - 2} \right)!}} = \dfrac{{3 \times 2 \times 1}}{{2 \times 1 \times 1!}} = \dfrac{6}{2} = 3 \\ {}^5{C_3};n = 5,c = 2 \\ {}^5{C_3} = \dfrac{{5!}}{{3!\left( {5 - 3} \right)!}} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1 \times 2!}} = \dfrac{{120}}{{12}} = 10 \\ \therefore No.of words = {}^3{C_2} \times {}^5{C_3} = 3 \times 10 = 30 \\ $Therefore, 30 words can be formed from the letters of the word DAUGHTER each containing 2 vowels and 3 consonants.

Note: A Permutation is arranging the objects in order. Combinations are the way of selecting the objects from a group of objects or collection. When the order of the objects does not matter then it should be considered as Combination and when the order matters then it should be considered as Permutation. Do not confuse using a combination, when required, instead of a permutation and vice-versa.

Your query why not permutation first ? As, you have to make words of length=$5$. And of these $5$, $2$ are vowels and $3$ consonants. Since, you have to first get those $2$ vowels and $3$ consonants to make the desired word. So first operation has to be combination(selection operation), which will select $2$ vowels out of $3$ vowels(A,E,U) and then you have to select 3 consonants out of $5$(D,G,H,T,R). And they need to be multiplied, as there can be many such combinations i.e $C(3,2)*C(5,3)$. Now that you have formed $5$ letter word. These letters can be arranged among themselves to make different words. Hence, you need to apply permutation(arrangement) i.e. $5!$, making final result= $C(3,2)*C(5,3)*5!$.