# 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?

360 | |

480 | |

720 | |

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?

810 | |

1440 | |

2880 | |

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?

72 | |

144 | |

360 | |

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?

5 | |

10 | |

15 | |

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?

120 | |

720 | |

4320 | |

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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