How many different ways can the letters of the word corporation be arranged so that the vowels always come together 1 point 810 1440 2880 50400?

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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.

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Exercise :: Permutation and Combination - General Questions

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2.

In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?

A. 360 B. 480 C. 720 D. 5040 E. None of these Answer: Option C Explanation: The word 'LEADING' has 7 different letters. When the vowels EAI are always together, they can be supposed to form one letter. Then, we have to arrange the letters LNDG (EAI). Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways. The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720. Video Explanation: https://youtu.be/WCEF3iW3H2c

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Page 2

Exercise :: Permutation and Combination - General Questions

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7.

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Answer: Option D Explanation: Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it. The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place. The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it. Required number of numbers = (1 x 5 x 4) = 20.

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Page 3

Exercise :: Permutation and Combination - General Questions

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13.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

A. 10080 B. 4989600 C. 120960 D. None of these Answer: Option C Explanation: In the word 'MATHEMATICS', we treat the vowels AEAI as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different. Number of ways of arranging these letters = 8! = 10080. (2!)(2!) Now, AEAI has 4 letters in which A occurs 2 times and the rest are different. Number of ways of arranging these letters = 4! = 12. 2! Required number of words = (10080 x 12) = 120960.