Answer

Verified

414.6k+ views

**Hint:**In this problem we need to find the number of $3$ letter words, each containing one vowel at least. For this, first, we will find a number of words each containing exactly one vowel. Then, we will find a number of words each containing exactly two vowels. In this problem, we have only two vowels $a$ and $e$. We will add the number of words containing exactly one vowel and the number of words containing exactly two vowels to get the total number of three-letter words. We will use the concept of combination and permutation in this problem.

**Complete step by step answer:**

We have two vowels $a,e$ and four consonants $b,c,d,f$. To find the number of three-letter words each containing one vowel at least, we will consider the following cases:

The case I: Exactly one vowel and two consonants.

In this case, one vowel out of two vowels can be selected in ${}^2{C_1} = 2$ ways. Two consonants out of four consonants can be selected in ${}^4{C_2} = \dfrac{{4 \times 3}}{{1 \times 2}} = \dfrac{{12}}{2} = 6$ ways.

All three letters (one vowel and two consonants) can arrange among themselves in $3! = 6$ ways. Therefore, the total number of three-letter words each containing exactly one vowel are $2 \times 6 \times 6 = 72$.

Case II: Exactly two vowels and one consonant.

Two vowels out of two vowels can be selected in ${}^2{C_2} = \dfrac{{2 \times 1}}{{1 \times 2}} = 1$ way. One consonant out of four consonants can be selected in ${}^4{C_1} = 4$ ways.

All three letters (one vowel and two consonants) can arrange among themselves in $3! = 6$ ways. Therefore, the total number of three-letter words each containing exactly two vowels is $1 \times 4 \times 6 = 24$. Now we will add the possibilities of the case I and case II to get the required number of words. Therefore, the total number of three-letter words each containing at least one vowel is $72 + 24 = 96$.

**Note:**

If we consider the order of objects then it is a permutation. If we do not consider the order of objects then it is a combination. ${}^n{C_r}$ gives the total number of ways of selecting $r$ objects out of $n$ objects. ${}^n{P_r}$ gives the total number of distinct arrangements when we arrange $r$ objects among $n$ objects. The number of permutations of $n$ distinct objects is $n!$.

Recently Updated Pages

Differentiate between Shortterm and Longterm adapt class 1 biology CBSE

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE