A) 4! x 4! | B) 5! x 5! |

C) 4! x 5! | D) 3! x 4! |

Explanation:

The word EDUCATION is a 9 letter word, with none of the letters repeating.

The vowels occupy 3rd,5th,7th and 8th position in the word and the remaining 5 positions are occupied by consonants

As the relative position of the vowels and consonants in any arrangement should remain the same as in the word EDUCATION, the vowels can occupy only the afore mentioned 4 places and the consonants can occupy1st,2nd,4th,6th and 9th positions.

The 4 vowels can be arranged in the 3rd,5th,7th and 8th position in 4! Ways.

Similarly, the 5 consonants can be arranged in1st,2nd,4th,6th and 9th position in5! Ways.

Hence, the total number of ways = 4! × 5!

A) 89,972 | B) 90,720 |

C) 72,000 | D) 81,000 |

Explanation:

The given word **HAPPYHOLI** has 9 letters

These 9 letters can e arranged in **9! ways.**

But here in the given word letters **H & P** are repeated twice each

Therefore, Number of ways these 9 letters can be arranged is

$\frac{\mathbf{9}\mathbf{!}}{\mathbf{2}\mathbf{!}\mathbf{}\mathbf{x}\mathbf{}\mathbf{2}\mathbf{!}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{9}\mathbf{}\mathbf{x}\mathbf{}\mathbf{8}\mathbf{}\mathbf{x}\mathbf{}\mathbf{7}\mathbf{}\mathbf{x}\mathbf{}\mathbf{6}\mathbf{}\mathbf{x}\mathbf{}\mathbf{5}\mathbf{}\mathbf{x}\mathbf{}\mathbf{4}\mathbf{}\mathbf{x}\mathbf{}\mathbf{3}}{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{90}\mathbf{,}\mathbf{720}\mathbf{}\mathbf{ways}\mathbf{.}$

A) 60 | B) 120 |

C) 240 | D) 30 |

Explanation:

Given letters are k, l, m, n, o = 5

number of letters to be in the words = 3

Total number of words that can be formed from these 5 letters taken 3 at a time without repetation of letters = $\mathbf{5}{\mathbf{P}}_{\mathbf{3}}\mathbf{}\mathbf{ways}\mathbf{.}$

$\Rightarrow 5{\mathrm{P}}_{3}=5\mathrm{x}4\mathrm{x}3=\mathbf{60}\mathbf{}\mathbf{words}\mathbf{.}$

A) 4320 | B) 4694 |

C) 4957 | D) 4871 |

Explanation:

Given Word is **PROMISE.**

Number** **of letters in the word PROMISE = **7**

Number of ways 7 letters can be arranged = **7! ways**

Number of Vowels in word PROMISE = **3 (O, I, E)**

Number of ways the vowels can be arranged that 3 Vowels come together = **5! x 3! ways**

Now, the number of ways of arrangements so that three vowels should not come together

=** 7! - (5! x 3!) ways** = 5040 - 720** = 4320.**

A) 2520 | B) 5040 |

C) 1260 | D) None |

Explanation:

The **7** letters word 'POVERTY' be arranged in $\mathbf{7}{\mathit{P}}_{\mathbf{7}}$ ways = 7! = 5040 ways.

A) 18 | B) 24 |

C) 36 | D) 32 |

Explanation:

Required Number of ways** = **$\mathbf{4}{\mathbf{C}}_{\mathbf{2}}\mathbf{}\mathbf{\times}\mathbf{}\mathbf{3}{\mathbf{C}}_{\mathbf{2}}\mathbf{}\mathbf{\times}\mathbf{}\mathbf{2}{\mathbf{C}}_{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{36}$** = 36**

A) 8640 | B) 720 |

C) 3620 | D) 4512 |

Explanation:

Let the 6 digits of the required 6 digit number be **abcdef**

Then, the number to be odd number the last digit must be odd digit i.e **3, 5, 7 or 9**

The first digit cannot be** ‘0’ **=> possible digits** = 3, 5, 7, 2, 6, 8**

Remaining 4 places can be of **6 x 5 x 4 x 3** ways

This can be easily understood by

Therefore, required number of ways = 6 x 6 x 5 x 4 x 3 x 4 = 36 x 20 x 12 = 720 x 12

= **8640 ways.**

A) 1260 | B) 2520 |

C) 5040 | D) 1080 |

Explanation:

The number of ways in which the letters of the word RAILINGS can be arranged such that R & S always come together is

Count R & S as only 1 space or letter so that RS or SR can be arranged => 7! x 2!

But in the word RAILINGS, I repeated for 2 times => 7! x 2!/2! = 7! ways = **5040 ways.**

A) 380 | B) 420 |

C) 441 | D) 400 |

Explanation:

Given total number of students in the class = 21

So each student will have 20 greeting cards to be send or receive (21 - 1(himself))

Therefore, the total number of greeting cards exchanged by the students = **20 x 21 = 420.**