Correct Answer: Description for Correct answer:
There are 8 letters in the word 'SOFTWARE including 3 vowels (O, A, E) and 5 consonants (S, F, T, W, R). Considering three vowels as one letter, we have six letters which can be arranged in \( \Large ^{6}P_{6}=6! \) ways. But corresponding to each way of these arrangements, the vowels can be put together in 3! ways. Required number of words = \( \Large 6! \times 3! \) = 4320
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word 'LEADING ' has 7 different letters. <br> when the vowels EAI are always together , they can be supposed to form one letter. <br> then , we have to arrange the letters LNDG (EAI) . <br> 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. <br> `therefore ` Required number of ways `=(120 xx6)= 720`
| 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
|
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.
|
| 2.
| From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
|
| A.
| 564
| | B.
| 645
| | C.
| 735
| | D.
| 756
|
Answer: Option D Explanation: We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).
| Required number of ways
| = (7C3 x 6C2) + (7C4 x 6C1) + (7C5)
| |
|
| =
|
| 7 x 6 x 5
| x
| 6 x 5
|
| + (7C3 x 6C1) + (7C2)
| | 3 x 2 x 1
| 2 x 1
|
| |
|
| = 525 +
|
| 7 x 6 x 5
| x 6
|
| +
|
| 7 x 6
|
| | 3 x 2 x 1
| 2 x 1
|
| |
| = (525 + 210 + 21)
| |
| = 756.
|
|
| 3.
| 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.
| 50400
|
Answer: Option D 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!
| = 2520.
| | 2!
|
Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged Required number of ways = (2520 x 20) = 50400.
|
| 4.
| In how many ways can the letters of the word 'LEADER' be arranged?
|
| A.
| 72
| | B.
| 144
| | C.
| 360
| | D.
| 720
|
Answer: Option C Explanation: The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.
| Required number of ways =
| 6!
| = 360.
| | (1!)(2!)(1!)(1!)(1!)
|
|
| 5.
| Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
|
| A.
| 210
| | B.
| 1050
| | C.
| 25200
| | D.
| 21400
|
Answer: Option C Explanation: Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4)
|
| = (7C3 x 4C2)
| |
|
| =
|
| 7 x 6 x 5
| x
| 4 x 3
|
| | 3 x 2 x 1
| 2 x 1
|
| |
| = 210.
|
Number of groups, each having 3 consonants and 2 vowels = 210. Each group contains 5 letters.
Number of ways of arranging
5 letters among themselves
| = 5!
| |
| = 5 x 4 x 3 x 2 x 1
| |
| = 120.
|
Required number of ways = (210 x 120) = 25200.
|
How many ways can the word studio be arranged in such a way that all vowels in the word come together?
Required number of ways = (120 x 6) = 720.
How many ways letter Organise can be arranged such that all vowels are come together?
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!
...
Permutation-and-Combination.. Can the letters of the word Leading be arranged in such a way that the vowels always come together?
 Required number of ways = (120 x 6) = 720. 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).
How many ways Word arrange can be arranged in which vowels are not together?
number of arrangements in which the vowels do not come together =5040−1440=3600 ways.
| 
