The number of 4-letter words, with or without meaning, each consisting of two vowels and t…

Question

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

Accepted Answer

From the word INCONSEQUENTIAL, let us first identify the vowels and consonants available.

The letters are:

$I,N,C,O,N,S,E,Q,U,E,N,T,I,A,L$

We have to form $4$-letter words containing:

exactly $2$ vowels

exactly $2$ consonants

no letter repeated

Since letters of the word may repeat in the original word, we count only distinct available letters for selection.

Distinct vowels: The vowels present are:

$A,E,I,O,U$

So, number of distinct vowels $=5$.

Distinct consonants: The consonants present are:

$N,C,S,Q,T,L$

So, number of distinct consonants $=6$.

Choose 2 vowels and 2 consonants: Number of ways to choose $2$ vowels from $5$:

${}^{5}{C}_{2}=10$

Number of ways to choose $2$ consonants from $6$:

${}^{6}{C}_{2}=15$

So, total selections of letters:

${}^{5}{C}_{2}	imes {}^{6}{C}_{2}=10	imes 15=150$

Arrange the 4 chosen letters: All 4 letters are distinct, so number of arrangements:

$4!=24$

Hence, total number of such 4-letter words is:

$150	imes 24=3600$

Therefore, the correct answer is:

$3600$

So, Option D is correct.

Solutions · 1