A) 9!/(2!)^{2}x3! | B) 9! x 2! x 3! |

C) 0 | D) None |

Explanation:

n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.

The letter pattern 'MESMERISE' consists of 10 letters of which there are 2M's, 3E's, 2S's and 1I and 1R.

Number of arrangements = $\frac{9!}{(2!{)}^{2}\times 3!}$

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.**

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**