A) 216 | B) 45360 |

C) 1260 | D) 43200 |

Explanation:

There are total 9 letters in the word COMMITTEE in which there are 2M's, 2T's, 2E's.

The number of ways in which 9 letters can be arranged = $\frac{9!}{2!\times 2!\times 2!}$ = 45360

There are 4 vowels O,I,E,E in the given word. If the four vowels always come together, taking them as one letter we have to arrange 5 + 1 = 6 letters which include 2Ms and 2Ts and this be done in $\frac{6!}{2!\times 2!}$ = 180 ways.

In which of 180 ways, the 4 vowels O,I,E,E remaining together can be arranged in $\frac{4!}{2!}$ = 12 ways.

The number of ways in which the four vowels always come together = 180 x 12 = 2160.

Hence, the required number of ways in which the four vowels do not come together = 45360 - 2160 = 43200

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.**

A) 851250 | B) 751210 |

C) 362880 | D) 907200 |

Explanation:

The number of letters in the given word CREATIVITY = **10**

Here T & I letters are repeated

=> Number of Words that can be formed from CREATIVITY = **10!/2!x2! **= 3628800/4 = **907200**

A) 14 × 13 × 13! | B) 15 × 14 × 14! |

C) 14 × 12 × 12! | D) 15 × 13 × 13! |

Explanation:

15! - 14! - 13!

= (15 × 14 × 13!) - (14 × 13!) - (13!)

= 13! (15 × 14 - 14 - 1)

= 13! (15 × 14 - 15)

= 13! x 15 (14 - 1)

= **15 × 13 × 13!**

A) 7920 | B) 74841 |

C) 14874 | D) 10213 |

Explanation:

ST candidates vacancies can be filled by ${}^{5}C_{3}$ ways = 10

Remaining vacancies are 5 that are to be filled by 12

=> ${}^{12}C_{5}$= (12x11x10x9x8)**/**(5x4x3x2x1) = 792

Total number of filling the vacancies = 10 x 792 = 7920