Question

In how many ways can two vowels and three consonants from the letters of the word ARTICLE?
$A)12$
$B)14$
$C)18$
$D)22$

Answer 1

First we have to define what the terms we need to solve the problem are.
Since the given question is to find the number of ways so we use to formula of the combination;
Combination is the number of ways to arrange or count the given problem.

Formula used:
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
n is the total amount and r is the object that needs to arrange into the different sets.

Complete step by step answer:
Since there are three vowels and four consonants in the word ARTICLE; that is AIE are the vowel letters and balance RTCL are the consonants letters; (consonants means non vowel letters in the alphabets)
Since there are three vowels and the number of ways of selecting the three vowels from the two require questions are $3{c_2}$(number of the ways)
So that AIE can be have $^3{c_2}$ways in selecting those two vowels for required question
Hence $^3{c_2} = 3$(three factorial divides two factorial)
Similarly, like same way since there are four consonants and the number of ways of selecting the four consonants from the three require question are $^4{c_3}$(number of the ways)
So that RTCL can be have $^4{c_3}$ ways in selecting those three vowels for required question
Hence $^4{c_3} = 4$(four factorial divides three factorial)
Hence combining the both results into multiplication to find the overall requirement is $4 \times 3 = 12$

So, the correct answer is “Option A”.

Note: ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$can be expressed as $^3{c_2} = \dfrac{{3!}}{{2!}}$
If the question is about the number of arrangements, then we use the formula of permutation which is
${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$ and $3{p_3} = 3!$