The letters of bottle are permuted in all possible ways how many of these words have all the vowels

In this section you can learn and practice Aptitude Questions based on "Permutation and Combination" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) with full confidence.

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You can easily solve all kind of Aptitude questions based on Permutation and Combination by practicing the objective type exercises given below, also get shortcut methods to solve Aptitude Permutation and Combination problems.

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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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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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.

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How many ways can the letters of the word BOTTLES be arranged such that both of the vowels are at the end?

Details and assumptions

The vowels in the word BOTTLES are O and E.

Among $5$ girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all $5$ girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other?

Hint: Treat the two girls as one person. This will help find the number of arrangements that have the girls seated together, then subtract the number from 5!, the total number of arrangements.

$10$ people including $A,B$ and $C$ are waiting in a line. How many distinct line-ups are there such that $A,B,$ and $C$ are not all adjacent?

Details and assumptions

$A,B$ and $C$ may be in any order as long as all three are not adjacent.

3 boys and 2 girls are about to be seated at a round table. If the 2 girls want to sit next to each other, find the number of ways seating these boys and girls.

(Note: since the table is round, we are considering two seating arrangements to be equivalent if they can match just by rotating.)

Mary has enrolled in $6$ courses: Chemistry, Physics, Math, English, French and Biology. She has one textbook for each course and wants to place them on a shelf. How many ways can she arrange the textbooks so that the English textbook is placed at any position to the left of the French textbook?

Hint: There are just as many permutations where the English textbook is to the left of the French textbook as there are permutation where the French textbook is to the left of the English textbook.