Permutation and Combination

Permutation and Combination

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 Question 1
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
Question 1 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.
 Question 2
In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
 A 810 B 1440 C 2880 D 5760
Question 2 Explanation:
In the word 'CORPORATION', we treat the vowels OOAIO as one letter. Thus, we have CRPRTN (OOAIO). This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different. Number of ways arranging these letters = 7!/2! = 2520. Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 5!/3! = 20 ways. Required number of ways = (2520 x 20) = 50400.
 Question 3
In how many ways can the letters of the word 'LEADER' be arranged?
 A 72 B 144 C 360 D 720
Question 3 Explanation:
The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R. Required number of ways =6!/((1!)(2!)(1!)(1!)(1!))= 360.
 Question 4
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?
 A 5 B 10 C 15 D 20
Question 4 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.
 Question 5
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
 A 120 B 720 C 4320 D 2160
Question 5 Explanation:
The word 'OPTICAL' contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5! = 120 ways. The vowels (OIA) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720.
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