How many ways can the letters of corporation be arranged so that the vowels are all together?

In how many different ways can the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together?

Nội dung chính

• In how many different ways can the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together?
• In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
• Solution(By Examveda Team)
• In how many different ways can the letters of the word "CORPORATION" be arranged so that all the vowels always come together?
• Answer (Detailed Solution Below)
• How many ways Corporation can be arranged if vowels are always together?
• How many ways can letters of the work article be arranged so that vowels always occupy odd places?
• How many ways word arrange can be arranged in which vowels are together?
• How many ways vowels can be arranged?

Answer

Hint: In the given question we are required to find out the number of arrangements of the word ‘CORPORATION’ so that the vowels present in the word always come together. The given question revolves around the concepts of permutations and combinations. We will first stack all the vowels together while arranging the letters of the given word and then arrange the remaining consonants of the word.

Complete step-by-step answer:
So, we are required to find the number of ways in which the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together.
So, we first stack all the vowels present in the word ‘CORPORATION’ separately. So, the vowels present in the word ‘CORPORATION’ are: ‘O’, ‘O’, ‘A’, ‘I’ and ‘O’.
So, we have three O’s , one I and one A.
Now, the remaining consonants in the word ‘CORPORATION’ are: C, R, P, R, T, N.
So, the number of consonants in the word ‘CORPORATION’ is $ 6 $ .
Now, we form a separate bag of the vowels and consider it to be one single entity. Then, we find the arrangements of the letters.
So, the number of entities to be arranged including the bag of vowels is $ 6 + 1 = 7 $ .
Now, the consonant R is repeated twice. So, the number of ways these seven entities can be arranged where one entity is repeated twice are $ \dfrac{{7!}}{{2!}} $ .
Also, there can also be arrangements in the bag of vowels. So, we have five vowels in the bag. But, the vowel O is repeated thrice.
Hence, the number of arrangements in the bag of vowels is $ \dfrac{{5!}}{{3!}} $ .
So, the total number of ways of arranging the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together are $ \dfrac{{7!}}{{2!}} \times \dfrac{{5!}}{{3!}} $ .
Substituting in the values of factorials, we get,
 $ \Rightarrow \dfrac{{5040}}{2} \times \dfrac{{120}}{6} $
Cancelling the common factors in numerator and denominator and simplifying the calculations, we get,
 $ \Rightarrow 50,400 $
So, the correct answer is “ $ \Rightarrow 50,400 $ ”.

Note: One should know about the principle rule of counting or the multiplication rule. Care should be taken while handling the calculations. Calculations should be verified once so as to be sure of the answer. One must know that the number of ways of arranging n things out of which r things are alike is $ \left( {\dfrac{{n!}}{{r!}}} \right) $ .

In how many different ways can the letters of the word "CORPORATION" be arranged so that all the vowels always come together?

1. 810
2. 1440
3. 2880
4. 50400

Answer (Detailed Solution Below)

Option 4 : 50400

Free

NIMCET 2020 Official Paper

120 Questions 480 Marks 120 Mins

Concept:

• The number of ways in which r objects can be arranged in n places is nPr = \(\rm \dfrac{n!}{(n - r)!}\).
• The number of ways in which all n objects can be arranged among themselves = nPn = n!.
• The number of ways in which n objects, out of which p, q, r etc. are of same type, can be arranged is: \(\rm \dfrac{n!}{p!\ q!\ r!\ ...}\).
• n! = 1 × 2 × 3 × ... × n.
• 0! = 1.

Calculation:

The word CORPORATION has 11 letters out of which 6 are consonants (CRPRTN) and 5 are vowels (OOAIO).

Considering the objects of the same type, the number of arrangements of these vowels will be \(\rm \dfrac{5!}{3!}\) = 20.

Since, the vowels have to be together, we can say that we have to arrange the groups (C), (R), (P), (R), (T), (N) and (OOAIO) among themselves.

Considering the objects of the same type, this can be done in \(\rm \dfrac{7!}{2!}\) = 2520 ways.

And, total number of arrangements of all the letters = [Number of arrangements of (C), (R), (P), (R), (T), (N) and (OOAIO)] × [Number of arrangements of (OOAIO)] = 20 × 2520 = 50400.

How many ways Corporation can be arranged if vowels are always together?

Required number of ways = (120 x 6) = 720. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

How many ways can letters of the work article be arranged so that vowels always occupy odd places?

We can form 144 words with the letters of the word ARTICLE where vowels occupy the even places and consonants the odd places.

How many ways word arrange can be arranged in which vowels are together?

The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.

How many ways vowels can be arranged?

The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720.

How many different ways CORPORATION be arranged so that vowels come together?

How many ways can the letters of the word CORPORATION be arranged so that vowels always occupy even places?

How many words can be formed from the word CORPORATION?

How many ways can the letters of the word Mussoorie be arranged so that the vowels always come together?