How many 5 letter words containing 3 vowels and 2 consonants can be formed using the letters of the word EQUATION?

Nội dung chính Show

How many five letter words containing 3 vowels and 2 consonants can be formed using the letters of the word ‘EQUATION’ so that the two consonants occur together?
The correct option is D Vowels (A,E,I,O,U) five, consonants (Q,T,N) three 3 vowels can be chosen in 5C3 ways 2 consonants can be chosen in 3C2 ways 2 consonants can be arranged in 2! Ways 3 vowels + 2 consonants bundle = 4 can be arranged in 4! Ways. ∴ required number words =5C33C22!4!
How many words each of 3 vowels and 2 consonants can be formed from the letters of the word equation?
What are some 5

A
720
B
1440
C
240
D
480

Solution:

There are 5 vowels and 3 consonants in the word EQUATION'. Three vowels out of 5 and 2 consonants out of 3 can be chosen 5C3×3C2 v..rays. So, there are 5C3×3 C2 groups each containing 3 consonants and two vowels. Now, each group contains 5 letters which are to be arranged in such a way that 2 consonants occur together. Considering 2 consonants as one letter, we have 4 letters which can be arranged in 4! ways. But, two consonants can be put together in 2! ,vays. Therefore, 5 letters in each group can be arranged in 4! x 2! ways.

∴ Required number of words = 5C3×3C2×4!×2!=1440

How many five letter words containing 3 vowels and 2 consonants can be formed using the letters of the word ‘EQUATION’ so that the two consonants occur together?

Answer

Verified

Hint: Here we need to find the number of different five letter words that can be formed using the letters in the given word. For that, we will first count the number of letters present in the given word and then we will count the number of vowels and number of consonants present in the word. Then we will find the number of ways to arrange these using the formula of permutation. After simplification, we will get the required value.Complete step-by-step answer:
The given word is ‘EQUATION’.
Total number of letters available in the word ‘EQUATION’ is equal to 8.
We can see from the word that there are five vowels in the word ‘EQUATION’ i.e. U, A, I, O, E and there are three consonants in the word ‘EQUATION’ i.e. Q, T, N.
We need to find the total number of words of five letters containing 3 vowels and 2 consonants. Therefore,
Number of ways to select three vowels \[ = {}^5{P_3}\]
Using this formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\], we get
Number of ways to select three vowels \[ = \dfrac{{5!}}{{\left( {5 - 3} \right)! \times 3!}}\]
On further simplification, we get
Number of ways to select three vowels \[ = \dfrac{{5!}}{{2! \times 3!}}\]
Computing the factorial, we get
Number of ways to select three vowels \[ = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 3 \times 2 \times 1}} = 5 \times 2 = 10\]
For consonants, we have
Number of ways to select two consonants \[ = {}^3{C_2}\]
Using this formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\], we get
Number of ways to select two consonants \[ = \dfrac{{3!}}{{\left( {3 - 2} \right)! \times 2!}}\]
On further simplification, we get
Number of ways to select two consonants \[ = \dfrac{{3!}}{{1! \times 2!}}\]
Computing the factorial, we get
Number of ways to select two consonants \[ = \dfrac{{3 \times 2 \times 1}}{{1 \times 2 \times 1}} = 3\]
Since, two consonants always occur together, so we will consider two consonants as one.
Number of ways to arrange these 5 letters considering two consonants as one \[ = 4! \times 2!\]
On further simplification, we get
Number of ways to arrange these 5 letters considering two consonants as one \[ = 4 \times 3 \times 2 \times 1 \times 2 \times 1 = 48\]
Hence, the number of different words formed by these 5 letters \[ = 10 \times 3 \times 48 = 1440\]
Therefore, the correct answer is equal to 1440.

Note: Here we have used the formula of combination and not permutation to get the required number of ways. Therefore we need to know the basic difference between the permutation and combination to avoid any mistakes. Permutation is used when we have to find the possible elements but the combination is used when we need to find the number of ways to select a number from the collection.

How many 5 letter words containing 3 vowels and 2 consonants can be formed using the letters of the word EQUATION?

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Solution

The correct option is D Vowels (A,E,I,O,U) five, consonants (Q,T,N) three 3 vowels can be chosen in 5C3 ways 2 consonants can be chosen in 3C2 ways 2 consonants can be arranged in 2! Ways 3 vowels + 2 consonants bundle = 4 can be arranged in 4! Ways. ∴ required number words =5C33C22!4!

How many 5

Number of different words formed by these 5 letters is 5! So the answer is C(5, 3)×C(3, 2)×5!

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word equation?

Therefore, the total number of words can be formed is = (⁴C3 X ⁴C2 X 5!) = 2880.

How many 5

Number of groups, each having 3 consonants and 2 vowels = 210. Each group contains 5 letters. = 5! = 120.

What are some 5

Wordle: 5-Letter Words With Three Vowels.

ABUSE..

ALONE..

ARGUE..

ARISE..

HOUSE..

JUICE..

MEDIA..

MOVIE..

How many 5 letter words containing 3 vowels and 2 consonants can be formed using the letters of the word EQUATION?

Solution:

How many five letter words containing 3 vowels and 2 consonants can be formed using the letters of the word ‘EQUATION’ so that the two consonants occur together?

The correct option is D Vowels (A,E,I,O,U) five, consonants (Q,T,N) three 3 vowels can be chosen in 5C3 ways 2 consonants can be chosen in 3C2 ways 2 consonants can be arranged in 2! Ways 3 vowels + 2 consonants bundle = 4 can be arranged in 4! Ways. ∴ required number words =5C33C22!4!

How many 5

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word equation?

How many 5

What are some 5

Bài Viết Liên Quan

Quảng Cáo

Có thể bạn quan tâm

Quảng cáo

Xem Nhiều

Quảng cáo

Chúng tôi

Điều khoản

Trợ giúp

Mạng xã hội