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a. 240

b. 360

c. 500

d. 720

**Answer:** a. 240

**Explanation:**

'DARKER' has 6 letters.

Tip:

We can arrange 'n' things in n! ways.

While arranging letters/things/numbers, if there are two same things or things get repeated twice, then we need to divide by 2!

If there are 3 same things or things get repeated thrice, then divide by 3! And so on..

Thus, we can arrange 6 letters in 6! ways.

But R gets repeated. There are 2 Rs. So divide by 2!

∴ Total ways = | 6! | = 360 |

2! |

Consider the 2 vowels (A and E) as one group.

We have 4 letters and 1 group = 5

We can arrange them in 5! Ways.

But again here R comes twice. So we will have | 5! |

2! |

So, Number of ways with vowels together = 2! x | 5! | = 120 |

2! |

a. 60

b. 70

c. 90

d. 147

**Answer:** c. 90

**Explanation:**

There are total 7 digits given - 2, 3, 4, 5, 6, 8 and 0

Between 500 and 1000 means from 501 to 999.

**All of them are 3 digit numbers.**

First digit | Second digit | Third digit |
---|---|---|

5, 6 or 8 i.e. 3 possibilities (1 digit gets used here) | Any one from 7-1 = 6 remaining digits i.e. 6 possibilities | Any one from 6-1 = 5 remaining digits i.e. 5 possibilities |

i. Two particular friends are definitely there

ii. Two particular members are definitely not there

a. 15 and 15

b. 15 and 360

c. 30 and 360

d. 360 and 360

**Answer:** a. 15 and 15

**Explanation:**

Tip:

SELECT = Combination = ^{n}C_{r} = | n! |

r!(n-r)! |

SELECT and ARRANGE = Permutation = ^{n}P_{r} = | n! |

(n-r)! |

When we include these 2 members, we are left with 4 - 2 = 2 spots in the group

Also, number of members which remain are 8 - 2 = 6

So now we need to select 2 people out of 6.

^{6}C_{2} = | 6! | = 15 = number of ways |

2!4! |

When we exclude these 2 members, we are left with 8 - 2 = 6 members

So now we need to select 4 people out of 6.

^{6}C_{4} = | 6! | = 15 = number of ways |

4!2! |

a. 14

b. 24

c. 48

d. 100

**Answer:** c. 48

**Explanation:**

**This is very easy to solve**

Say out of 8, Priyanka chooses one way to go from London to Delhi.

Now, From Delhi to Berlin she has 6 routes i.e. 6 options

Similarly, for 2^{nd} route between London and Delhi, Priyanka will again have 6 options from Delhi to Berlin.

This is true for all 8 routes between London and Delhi.

So, answer = **8 x 6 = 48 possible ways to travel from London to Berlin via Delhi**

a. 1/2

b. 13

c. 22

d. 35

**Answer:** d. 35

**Explanation:**

Leader has to be from cricketers or wrestlers i.e. from 12 cricketers and 10 wrestlers.

So, there are only 12 cricketers + 10 wrestlers **= 22 ways to select a leader**

Going Further:

**If it is said that -** The organizer of the group wants any one person to be leader of the group, then

The number of ways simply would be 35.

Because there are 35 people and anyone can be selected as leader.